Given a triangle ABC with an inscribed circle with a center O:

enter image description here

we create three triangles from an inscribed circle, $\triangle OA_1B_1$, $\triangle OB_1C_1$, $\triangle OC_1A_1$. Then, we define three points $O_1, O_2, O_3$ which are the centers of the circles that circumscribe around $\triangle OB_1C_1$, $\triangle OC_1A_1$, $\triangle OA_1B_1$ respectively.

We aim to find the area of $\triangle O_1O_2O_3$, given that the area of $\triangle ABC$ is 24.

I thought $\triangle O_1O_2O_3$ would perhaps be the equilateral triangle that inscribes the circle $O$, as $\angle A_1OB_1$, $\angle B_1OC_1$, $\angle C_1OA_1$ would be obtuse, I cannot either back my assumption up or derive a meaningful result from my proposition.

Any insights on finding the area of the desired triangle would be helpful.


Observe that, points $O$, $C_{1}$, $A$ and $B_{1}$ are concyclic. Moreover, they all lie on the circle with diameter $OA$. Thereafter $O_{1}$ is the midpoint of $OA$. Arguing similarly will show that $O_{2}$ and $O_{3}$ are the midpoints of $OB$ and $OC$. Now proceed.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.