Problem
GEO-B3-M06-P012 Symmedians via Trig Ceva
#12
★★★☆☆ Level 3 of 5
Using trig Ceva, prove that the three symmedians of a triangle are concurrent.
Inspired by Prasolov trigonometric geometry method
C. Hint 1. Symmedians are isogonal to medians.
D. Hint 2. Medians are concurrent, so apply the isogonal-cevian result.
The medians of a triangle are concurrent at the centroid. By the previous problem, the isogonal cevians of a concurrent triple are also concurrent.
The isogonal images of the medians are the symmedians. Therefore the three symmedians are concurrent. Their point is the Lemoine point.
This is more conceptual than the ordinary Ceva proof with squared side ratios.