MrMineev

The Anchor Point Lemma

Daniel Alex Mineev, RANDOM__USER

1 Theory

The Anchor Point Lemma is a series of configurations surrounding circles passing through D with well-defined intersections with sides of a triangle.

1.: (Anchor Point Lemma) Given a triangle $△ABC$ , let $D$ be an arbitrary point on $BC$ , then let $DE$ and $DF$ be parallel to $AC$ and $AB$ respectively. Let $(AEF )$ intersect $(ABC )$ at $G$ , let $GD$ intersect $(ABC )$ at $X$ . Prove, that if $M$ is the midpoint of $AC$ , then $A,X$ and $M$ are colinear.
2.: (Generalized Anchor Point Lemma) Let $D$ be an arbitrary point on $BC$ in $△ABC$ . Let there be two fixed directions $l1$ and $l2$ . Let $E$ and $F$ be the intersection of two lines through $D$ parallel to $l1$ and $l2$ with $AB$ and $AC$ . Let $G$ be the intersection of $(AF E)$ and $(ABC )$ . Prove that $DG$ passes through a constant point X on $(ABC )$ .
Also prove that the circles $(AF E)$ are all coaxial and the second constant point is the intersection of the circle with the line isogonal to $AX$ in $∠CAB$ with $(AF E )$ .

Remark 1.1 The circle (AGD) passes through a fixed point K on BC such that AK is isogonal to AX. If P = CE ∩ BF then the line DP passes through a fixed point.
3.: (Advanced Anchor Point Lemma) Let $P$ be an arbitrary fixed point and $X$ an abitrary fixed point on $(ABC )$ . Let $D$ be an abritrary point on $BC$ . Let $P D$ intersect $AC$ at $E$ . Let $XD$ intersect $(ABC )$ at $G$ . Let $(AEG )$ intersect $AB$ a second time at $F$ . Prove that the line $DF$ passes through a constant point $Q$ as $D$ moves on $BC$ , and that $(AEF )$ passes through a fixed point $W$ .

Remark 1.2 Similarly the to the previous lemma if P = CE ∩ BF then the line DP passes through a fixed point.

Note: More details about this method can be found at https : ∕∕mrmineev.com∕articles∕anchor - point∕main.html

2 More Advanced Theory

Let Anchor_P^ABC(E) be the Anchor Point conjugate of point P where E is the parameter. Notice,

1.: Anchor_E^△ABC(AC) is a line which passes through C and the intersection of AE with the line through B parallel to AC.
2.: Anchor_E^△ABC(AB) is a line which passes through B and the intersection of AE with the line through B parallel to AC.
3.: Anchor_(ABC)^△ABC(P) is a line parallel to AC which passes through the intersection of the line through P parallel to AB with BC.
4.: Let be an arbitrary line in the plane then Anchor_E^△ABC() is a line which passes through ∩ BC.

In addition to these properties the cross-ratio is preserved.

3 Problems

1.: (The Shooting Lemma) Consider the chord $BC$ in the circle $Ω$ . Let the circle $ω$ touch $BC$ at a point $D$ and the circle $Ω$ at a point $E$ . Prove that the line $DE$ passes through $M$ , the middle of the larger arc $⌢ BC$ .
2.: (Sharky-Devil Config) Let $(I)$ be the incircle of $△ABC$ , let $D, E,F$ be the tangency points of $(I)$ with $BC, AB$ and $AC$ , respectively. Let $G$ be the second intersection of $(AFE )$ with $(ABC )$ . Let $S$ be the midpoint of the arc $BC$ . Prove that $S,D$ and $G$ are colinear.
3.: (USA TST 2008 P7) Let ABC be a triangle with G as its centroid. Let P be a variable point on segment BC. Points Q and R lie on sides AC and AB respectively, such that PQ ∥ AB and PR ∥ AC. Prove that, as P varies along segment BC, the circumcircle of triangle AQR passes through a fixed point X such that ∠BAG = ∠CAX.
4.: (USA TST 2012 P1) In acute triangle ABC, ∠A < ∠B and ∠A < ∠C. Let P be a variable point on side BC. Points D and E lie on sides AB and AC, respectively, such that BP = PD and CP = PE. Prove that as P moves along side BC, the circumcircle of triangle ADE passes through a fixed point other than A.
5.: (ELMO 2013 Shortlist G3) Given △ABC, a point D lies on line BC. The circumcircle of ABD meets AC at F (other than A), and the circumcircle of ADC meets AB at E (other than A). Prove that as D varies, the circumcircle of AEF always passes through a fixed point other than A, and that this point lies on the median from A to BC.
6.: (APMO 2012 P4) Let ABC be an acute triangle. Denote by D the foot of the perpendicular line drawn from the point A to the side BC, by M the midpoint of BC, and by H the orthocenter of ABC. Let E be the point of intersection of the circumcircle Γ of the triangle ABC and the half line MH, and F be the point of intersection (other than E) of the line ED and the circle Γ. Prove that = must hold.
7.: (based on Moscow Mathematical Olympiad 2015) Let △ABC be an isoseles triangle. Let P and Q be points on sides AB and AC, respectively, such that BP = AQ. Let X be a point on BC such that PX = PB and T be the second intersection of (APQ) with (ABC). Prove that ∠ATX = 90.
8.: (IMO Shortlist 2016 G2) Let ABC be a triangle with circumcircle Γ and incenter I and let M be the midpoint of BC. The points D, E, F are selected on sides BC, CA, AB such that ID ⊥BC, IE ⊥AI, and IF ⊥AI. Suppose that the circumcircle of △AEF intersects Γ at a point X other than A. Prove that lines XD and AM meet on Γ.
9.: (BMO Shortlist 2017 G5) Let ABC be an acute angled triangle with orthocenter H. centroid G and circumcircle ω. Let D and M respectively be the intersection of lines AH and AG with side BC. Rays MH and DG interect ω again at P and Q respectively. Prove that PD and QM intersect on ω.
10.: (APMO 2022 P2) Let ABC be a right triangle with ∠B = 90^∘. Point D lies on the line CB such that B is between D and C. Let E be the midpoint of AD and let F be the seconf intersection point of the circumcircle of △ACD and the circumcircle of △BDE. Prove that as D varies, the line EF passes through a fixed point.
Let X be a point on the arc AB of Γ not containing C, such that ∠AXH = ∠AFH. Let K be the circumcenter of triangle XIA. Prove that the lines AO and KI meet on Γ.
11.: (IMO Shortlist 2005 G5) Let △ABC be an acute-angled triangle with AB ⁄= AC. Let H be the orthocenter of triangle ABC, and let M be the midpoint of the side BC. Let D be a point on the side AB and E a point on the side AC such that AE = AD and the points D, H, E are on the same line. Prove that the line HM is perpendicular to the common chord of the circumscribed circles of triangle △ABC and triangle △ADE.
12.: (by @SBYT) Given a triangle △ABC with an incenter I, let D be a point on segment BC. Let the perpendicular line from D to CI meet BI at E and the perpendicular line from D to BI meet CI at F. Let the perpendicular line from D to BC meet EF at G. Prove that the circles (ABC) and the circle centered at G with radius GD are tangent.

4 Some of my Own

1.: (AoPS) Let $D$ be an arbitrary point on the side $BC$ of triangle $△ABC$ . Let $E$ and $F$ be the intersections of the lines through $D$ , parallel to $AC$ and $AB$ , with $AB$ and $AC$ , respectively. Let $G$ be the intersection point of the circumcircle of triangle $AF E$ with the circumcircle of triangle $ABC$ . Let $M$ be the midpoint of $BC$ , and let $X$ be the second intersection point of line $AM$ with the circumcircle of triangle $ABC$ . Prove that lines $EF$ , $AD$ , and the tangent to the circumcircle of triangle $XM G$ at point $M$ are concurrent.
2.: (AoPS) Let $D$ be an arbitrary point on the side $BC$ in a triangle $△ABC$ . Let $E$ and $F$ be the intersection of the lines parallel to $AC$ and $AB$ through $D$ with $AB$ and $AC$ . Let $G$ be the intersection of $(AF E )$ with $(ABC )$ . Let $M$ be the midpoint of $BC$ and $X$ the intersection of $AM$ with $(ABC )$ . Let $H$ be the intersection of $(XM G)$ with $BC$ . Prove that $EF$ is parallel to $AH$ .
3.: (AoPS) Let $D$ be an arbitrary point on the side $BC$ in a triangle $△ABC$ . Let $E$ and $F$ be the intersection of the lines parallel to $AC$ and $AB$ through $D$ with $AB$ and $AC$ . Let $G$ be the intersection of $(AF E )$ with $(ABC )$ . Let $M$ be the midpoint of $BC$ and $X$ the intersection of $AM$ with $(ABC )$ . Let $J$ be the intersection of $(XF G )$ with $AC$ . Prove that $XB$ , $AD$ and $JM$ are concurrent at $P$ .

5 A bit more theory...

The Anchor Point Conjugate Theorem can be reformulated as,

Theorem 5.1 Fix a point $F ∈ (ABC )$ and an arbitrary point $D$ in the plane. Let $ω$ be a circle passing through $A$ and $D$ . Let $X$ and $Y$ be the second intersection points of $ω$ with $AC$ and $AB$ , respectively. Let $Q$ be the second intersection of $ω$ with the circumcircle $(ABC )$ , and define

P = F Q ∩BC.

Then, as $ω$ varies, the lines $P X$ and $PY$ each pass through a fixed point.

Let these two points be called the Evil Anchor Point conjugates of D in △ABC with respect to F, which will be denoted as Evil_F,W^△ABC(D) were W is either B or C depending on whether it is the point which lies on PX or PY .

1.: (Evil Characterization) Let (ABD) meet lines AC and AF at X and Y , respectively, then Evil_F,B^△ABC(D) = BX ∩ DY .
2.: (AoPS) Prove that, $( |{ D ∈ Evil△FA,BBC (D )Evil△FC,BBA (D ) D ∈ Evil△BAC (D )Evil△CAB (D ) |( F△,AABC F△,ABAC D ∈ EvilF,C (D )EvilF,C (D )$

6 Extra (Unsolved)

1.: (USEMO 2025 P2) Let ABC be a fixed triangle with circumcircle ω. Consider P a variable point inside ABC. Ray BP meets side AC at Y while ray CP meets side AB at X. Let Q be the second intersection of ω and the circumcircle of triangle AXY . Let K be the second interesction of ray AP and ω.
Prove that as P varies, the circumcircles of triangle QPK all have a common radical center.

7 Proposed Solutions to Some Problems

Problem 7.1 (Sharky-Devil Configuration)

Let $(I)$ be the incircle of $△ABC$ , let $D, E,F$ be the tangency points of $(I)$ with $BC, AB$ and $AC$ , respectively. Let $G$ be the second intersection of $(AF E)$ with $(ABC )$ . Let $S$ be the midpoint of the arc $BC$ . Prove that $S,D$ and $G$ are colinear.

Proof. Let us introduce $M$ the midpoint of $BC$ and $W$ the midpoint of the larger arc $BC$ . Then, let $X$ and $Y$ be the intersections of the lines through $M$ parallel to $DE$ and $DF$ with $AB$ and $AC$ , respectively. By the Generalized Anchor Point Lemma all that is left to prove is that $AW XY$ is cyclic.

Notice, since,

∡W XY = ∡W AC

∡XY W = ∡XAW

which implies that $∠W AC = 180- ∠BAW$ , however this is only true for $W$ being the midpoint of the larger arrc $BC$ . Thus $W AXY$ is cyclic which proves one of the properties of the Sharky-Devil point. (Amusingly $I$ lies on this circle as well due to $I ∈ (AF E)$ and $I$ lying on the angle bisector of $∠CAB$ ). ■

Problem 7.2 (USA TST 2012 P1)

In acute triangle ABC, ∠A < ∠B and ∠A < ∠C. Let P be a variable point on side BC. Points D and E lie on sides AB and AC, respectively, such that BP = PD and CP = PE. Prove that as P moves along side BC, the circumcircle of triangle ADE passes through a fixed point other than A.

Proof. Since as we move $P$ the lines $EP$ and $P D$ are parallel to two fixed directions, thus by the Generalized Anchor Point Lemma it must be that $(AED )$ passes through a fixed point. ■

Problem 7.3 (ELMO 2013 Shortlist G3)

n △ABC, a point D lies on line BC. The circumcircle of ABD meets AC at F (other than A), and the circumcircle of ADC meets AB at E (other than A). Prove that as D varies, the circumcircle of AEF always passes through a fixed point other than A, and that this point lies on the median from A to BC.

Proof. Notice, $FD$ and $DE$ point in constant directions, since $∠CDF = ∠A = ∠EDB$ . Thus, by the Generalized Anchor Point Lemma all we need to do is show for one position of $D$ that $(AF E)$ passes through some fixed point on the median. Let us fix $D$ to be the foot of the altitude from $A$ to $BC$ . Let $S$ be the intersection of the symmedian from $A$ with $(ABC )$ , then, (it is well known, however the proof is outlined below)

Lemma 7.1 $S,D, Q$ are colinear.

indeed, since $ABSC$ is harmonic, by projecting from $T$ it must be that $QBCX$ is harmonic, consequently projecting from $D$ we obtain that $Q$ goes to a point $W$ on $(ABC )$ such that $ABCW$ is harmonic, thus $W = S$ , thus $Q, D,S$ are colinear. $□$

Now, by the Generalized Anchor Point Lemma since $S,D, Q$ are colinear, it must be that $(AEF )$ passes through a fixed point lying on the isogonal line to $AS$ in $∠CAB$ which is the median. ■

Problem 7.4 (IMO Shortlist 2016 G2)

Let ABC be a triangle with circumcircle Γ and incenter I and let M be the midpoint of BC. The points D, E, F are selected on sides BC, CA, AB such that ID ⊥BC, IE ⊥AI, and IF ⊥AI. Suppose that the circumcircle of △AEF intersects Γ at a point X other than A. Prove that lines XD and AM meet on Γ.

Proof. Let K = AM ∩ (ABC). Let P and Q be the intersections of parallel lines through D with respect to AB and AC with AC and AB, respectively.

Let L = MX ∩ (ABC), R = AL∩BC and G = PQ∩AD. From the Anchor Point lemma we know that GM ∥ AL, however it is a well known lemma that M,I and G are collinear. Thus we see that MI ∥ AR and the reflection of D over I lies on AR, consequntly it must be that M is the midpoint of RD.

Concluding by Butterly theorem we see that K,D and X are collinear. ■

Problem 7.5 (SBYT)

Given a triangle △ABC with an incenter I, let D be a point on segment BC. Let the perpendicular line from D to CI meet BI at E and the perpendicular line from D to BI meet CI at F. Let the perpendicular line from D to BC meet EF at G. Prove that the circles (ABC) and the circle centered at G with radius GD are tangent.

Proof. Let P = DE ∩ AC, Q = DF ∩ AB and T = DW ∩ (ABC), then notice that,

Lemma 7.2 AEQFPI is cyclic

Indeed, since P and Q are the reflections of D over CI and BI we know that,

∠F PD = ∠FDP = ∠EQF

and,

180- 2∠QDP ∠P F D = -------------= ∠A 2

consequently APQEF is cyclic. By the Anchor Point Lemma we know that since as D moves PD and QD have constant directions and when D is the projection of I onto BC by Sharky-Devil we know that TD passes through W the midpoint of the arc BC. Thus TD always passes through W, but Anchor Point then tells us that the circle passes through a constant point lying on the isogonal conjugate of AW in ∠CAB which implies that since I lies on the circle when D is the projection that I always lies on the circle. □

Thus, since TD passes through W all we have to show to prove that (GD) is tangent to (ABC) is that T,G,O are collinear, where O is the circumcenter of △ABC.

If D moves with degree 1 on BC then T moves with degree 2, similarly points F and E move with degree 1. Thus, the line EF moves with degree 2 and the line l (perpendicular from D to BC) moves with degree 1. Thus, G moves with degree 3. Thus, to show that T,G,O are collinear it sufficies to consider 2 + 3 + 1 = 6 cases.

Consider D being equal to B, C, the midpoint of BC, the projection of I onto BC (then one needs to prove that the reflection of D over G lies on (APQ) which is trivial by angle chase), AI ∩ BC and finally ∞_BC.

Consequently since T,G,O are collinear and T,D,W are collinear it must be that (GD) is tangent to (ABC). ■

Problem 7.6 (APMO 2012 P4)

Let ABC be an acute triangle. Denote by D the foot of the perpendicular line drawn from the point A to the side BC, by M the midpoint of BC, and by H the orthocenter of ABC. Let E be the point of intersection of the circumcircle Γ of the triangle ABC and the half line MH, and F be the point of intersection (other than E) of the line ED and the circle Γ. Prove that BCFF- = AABC- must hold.

Proof. Notice, it is well know that T is the A-Queue point of △ABC. Thus, if X and Y are the feet of the altitudes from B and C it must be that AXHY T is cyclic. Notice, by ELMO 2013 Shortlist G3 we obtain that since AY DC and AXDB are cyclic, consequently we know that (AXY ) passes through a fixed point on the median, by Generalized Anchor Point we know that DT passes through the point F on (ABC) such that AF is the symmedian, thus ABFC is harmonic and the problem statement follows. ■

Remark 7.3 The problem can also be solved by noticing that by Generalized Anchor Point since ATMD is cyclic (since ∠ATM = ∠ADM = 90) it must be that TD passes through a point on (ABC) isogonal to AM w.r.t. ∠CAB.

Problem 7.7 (APMO 2022 P2)

Let ABC be a right triangle with ∠B = 90^∘. Point D lies on the line CB such that B is between D and C. Let E be the midpoint of AD and let F be the seconf intersection point of the circumcircle of △ACD and the circumcircle of △BDE. Prove that as D varies, the line EF passes through a fixed point.

Proof. Let K be such that CDAK is a trapezoid.

Let G be the reflection of C over B. Then let P = Anchor_K^△ABC(G) then is known that the Anchor Point conjugates of points on CD will lie on CK, thus P ∈ CK. However, it is also known that the Anchor Point transformation preserves cross-ratios which in the case of lines, implies the preservation of ratios thus since,

{ △ABC AnchorK (D) = K Anchor△KABC (C) = C

thus we know that from the preservation of ratios that DK ∥ GP, however AG ∥ DK, thus G,A and P are collinear and K,E,G are collinear.

Then let N = PB ∩ AC, trivially N ∈ KG, however now due to the definition of the Anchor Point conjugate K ∈ NF which implies that G ∈ EF. ■

Problem 7.8 (IMO Shortlist 2005 G5)

Let △ABC be an acute-angled triangle with AB ⁄= AC. Let H be the orthocenter of triangle ABC, and let M be the midpoint of the side BC. Let D be a point on the side AB and E a point on the side AC such that AE = AD and the points D, H, E are on the same line. Prove that the line HM is perpendicular to the common chord of the circumscribed circles of triangle △ABC and triangle △ADE.

Proof. Let X = (AFB)∩AC and F = AH ∩(AED). Since AF is the isogonal line of AA′ (where A′ is the antipode), then we know that the Evil Anchor Point conjugates of F with respect to A′ are some infinity points.

Thus, by the Evil Anchor Point characterization theorem we must only show that BX ∥ EM, which is equivalent to E is the midpoint of CX. Notice,

Lemma 7.4 △DFB ~△EFX

Indeed since ∠EXF = ∠ABF and ∠XEF = ∠FDB. □

Similarly,

Lemma 7.5 △CEH ~△BDH

Indeed, since ∠CEH = ∠HDB and ∠ACH = ∠ABH. □

Consequently we know that,

( |{ BDDF- = EEXF- CE- = DB- |( EEHH- DEHF- HD = FD

the desired result now trivially follows,

EX = BD--⋅EF = BD ⋅ EH-= EH ⋅ CE-= CE DF HD EH

thus CE = EX. ■