This is an old open problem by Oai Thanh Đào. Here I give a proof.
Consider a triangle ABC. Let D be a point inside triangle ABC. Let EFG be the cevian triangle of point D. On segments BF, CF, CG, AG, AE, BE erect similar isosceles triangles BFH, CFI, CGJ, AGK, AEL and BEM, respectively. Let N be the intersection of HI, LM. Similarly, let O be the intersection of HI, KJ and P the intersection of KJ, ML. Then, AP, BN, CO concur.
Proof (click on the image to have a better view).
Related topics:
A family of perspectors associated to cevians
