Chicken claw theorem: the distance from the intersection of the bisector of a triangle and its circumscribed circle to the other two vertices is equal to the distance from the center to the edge center.

Chicken claw theorem means that if the center of △ABC is I, the lateral center in ∠A is J, and the extension line of AI intersects the triangle and circumscribes K, then KI=KJ=KB=KC. The graph composed of KI, KJ, KB and KC looks like a chicken claw, so it is called the chicken claw theorem.

Inverse principle

Let the bisector of ∠BAC in △ABC intersect the circumscribed circle of △ABC at k, and the intercept KI=KB=KJ on AK and the extension line, where I is inside △ABC and J is outside △ABC. Then point I is the heart of △ABC, and point J is the lateral heart of △ABC.

Proof: The inverse theorem of this theorem can be easily proved in the same way.

Take the inner I' and the paracentric J' of △ABC, and according to the theorem, KB=KC=KI'=KJ'

Again: KB=KI=KJ

∴I and I overlap, and J and J overlap.

That is, I and J are heart and lateral heart respectively.