$D$, $E$ and $F$ are points respectively on sides $BC$, $AB$ and $AC$ such that $AE = CD$ and $AF = BD$. $(BDE) \cap (CDF) = P \not\equiv D$. Prove that $PD$ passes through the excenter of $\triangle ABC$ in $\angle A$.

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I have noticed that if $K$ is the midpoint of arc $BC$ not containing point $A$ and $EF \cap KD = H$ then $B, E, H, K$ and $C, F, H, K$ are concyclic.

$\implies \angle AEF = \angle BKD$ and $\angle AFE = \angle CKD$.

Not sure what is next though.

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  • $\begingroup$ I have posted a proof that sadly doesn't incorporate your observation, it's really cool. How did you show that the points $B,E,H,K$ are concyclic? (I proved it by first proving $\angle AEF = \angle BKD$, but you seem to have done it the other way round, which is quite interesting). $\endgroup$ – Josef E. Greilhuber Sep 21 '19 at 20:16

The excenter is the radical center of the circumcircle $(ABC)$ and the two circles $(BDE)$ and $(CDF)$, thus the radical axis $DP$ of the latter two circles has to pass through it.

To prove this, let $N$ be the midpoint of the arc $AB$ of the circumcircle which contains $C$. Then $NA = NB$ holds, and by the inscribed angle theorem, $\angle DBN = \angle FAN$. Together with $DB = FA$, this implies that the triangles $DBN$ and $FAN$ are congruent (and have the same orientation), and hence $\angle BND = \angle ANF$. This yields, together with the inscribed angle theorem, $$\angle FCD = \angle ACD = \angle AND = \angle ANF + \angle FND - \angle BND = \angle FND,$$ and thus $N$ lies on the circle $(CDF)$. Therefore, the radical axis of $(ABC)$ and $(CDF)$ is the line $CN$, which is the exterior angle bisector in $C$ (this is well known, but also immediate from the inscribed angle theorem, since $AN=BN$). Analogously, the radical axis of $(ABC)$ and $(BDE)$ is the exterior angle bisector in $B$, and thus the radical center of the three circles is the excenter, as claimed.

Sketch of the first claim's proof.

  • $\begingroup$ That was a good solution, an excellent solution actually. Thanks for that. $\endgroup$ – Lê Thành Đạt Sep 22 '19 at 2:28

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