Concurrency of Lines Involving Altitudes and Circumcenters in a Triangle
by JackMinhHieu, Jul 14, 2025, 6:53 AM
Hi everyone,
I recently came across an interesting geometry problem that I'd like to share. It involves a triangle inscribed in a circle, altitudes, points on arcs, and a surprising concurrency involving circumcenters. Here's the problem:
Problem:
Let ABC be an acute triangle inscribed in a circle (O). Let the altitudes AD, BE, CF intersect at the orthocenter H.
Points M and N lie on the minor arcs AB and AC of circle (O), respectively, such that MN // AB.
Let I be the circumcenter of triangle NEC, and J be the circumcenter of triangle MFB.
Prove that the lines OD, BI, and CJ are concurrent.
I find the configuration quite elegant, and I'm looking for different ways to approach the problem — whether it's synthetic, coordinate, vector-based, or inversion.
Any ideas, hints, or full solutions are appreciated. Thank you!
I recently came across an interesting geometry problem that I'd like to share. It involves a triangle inscribed in a circle, altitudes, points on arcs, and a surprising concurrency involving circumcenters. Here's the problem:
Problem:
Let ABC be an acute triangle inscribed in a circle (O). Let the altitudes AD, BE, CF intersect at the orthocenter H.
Points M and N lie on the minor arcs AB and AC of circle (O), respectively, such that MN // AB.
Let I be the circumcenter of triangle NEC, and J be the circumcenter of triangle MFB.
Prove that the lines OD, BI, and CJ are concurrent.
I find the configuration quite elegant, and I'm looking for different ways to approach the problem — whether it's synthetic, coordinate, vector-based, or inversion.
Any ideas, hints, or full solutions are appreciated. Thank you!