Nine-Point Circles

Aryth

An arbitrary point $P$ is chosen in the plane of $\triangle ABC$ . Prove that the circumcircles of the medial triangles of $\triangle ABC$ , $\triangle PAB$ , $\triangle PBC$ , and $\triangle PCA$ are concurrent.

**Posted:** Mon Aug 18, 2008 7:57 am

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plane geometry · Riemann Hypothesis Offline Joined: 06 May 2008 Posts: 468

Let ⊙O1 ⊙02 meet at X
∠EXJ=∠EKH+∠HGJ=∠BPA=∠EFJ
=> E,F,X,J concyclic
∠FXE+∠EXJ+∠GXJ=∠EAF+∠BPA+∠GBJ=180+∠ABC=180+∠FKG
∠FXG=360-180-∠FKG=180-∠FKG
=> X,F,K,G concyclic
done！

**Posted:** Tue Aug 19, 2008 7:48 am

CatalystOfNostalgia

Here is an extension that I thought of...given the point X on the nine-point circle of ABC, construct the locus of all possible P.

**Posted:** Wed Aug 20, 2008 4:07 pm

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Analogies are like forgetful functors - you always lose information about the source.