Jacobi’s Theorem (A Huge Trig Bash)

So here is Jacobi’s Theorem in all it’s glory: Let be a triangle and let be points in its plane such that , , and . Then the lines are concurrent.

Jacobi’s Theorem is pretty interesting because it trivializes the existence of points such as the First Napoleon Point, First Fermat Point, and in general, the Kiepert points. They are actually more simple cases of Jacobi’s Theorem because the triangles erected are all isosceles.

Proof:
Like the title suggests, the proof is a huge trig bash. For the sake of clarity, let , , . We know that . Rearranging, we obtain . Similarly, . Thus . We now apply Law of Sines again on to compute . Thus . We are now done by Trig Ceva.

Alternate Proof:
Use the fact that concur at . We obtain . We also have similar equations for and . Multiply them together and use the fact that the external segments are isogonal to get equal angles and you’re done.

Problem: Prove Kariya’s Theorem, which states that if are the tangency points of the incircle of triangle with the sides and if points lie on lines with being the incenter of satisfying and all lie towards the interior or exterior of the triangle, then the lines are concurrent.