Midcircles!!!

For students of class 6-8 (age 12 to 14)
ShurjoA
Posts:8
Joined:Thu Dec 03, 2020 12:02 pm
Midcircles!!!

Unread post by ShurjoA » Thu Apr 01, 2021 10:10 pm

Hello!

This topic may be unrelated, but I just needed to get this out there. I have "discovered" something, which I have named midcircles.
I was just proving some theorems from an old book. Here are the theorems:
The internal angle bisectors of $\angle ABC$ and $\angle ACB$ intersect at $D.$ The external angle bisectors of $\angle ABC$ and $\angle ACB$ intersect at $E.$
incenter and excenter.JPG
incenter and excenter.JPG (23.52KiB)Viewed 3245 times
1. Prove that, $\angle BDC = 90^o + \frac{1}{2} \times \angle BAC.$
Proof:
$\angle ABC + \angle ACB + \angle BAC = 180^o$
$\angle ABC = 2\angle CBD, \angle ACB = 2\angle BCD$
$2\angle BCD + 2\angle CBD + \angle BAC = 180^o$
$\angle BCD + \angle CBD = 90^o - \frac{1}{2} \angle BAC$
$\angle BCD + \angle BDC + \angle CBD = 180^o$
$90^o - \frac{1}{2} \angle BAC + \angle BDC = 180^o$
$\angle BDC = 90^o + \frac{1}{2} \times \angle BAC. \blacksquare$
2. Prove that, $\angle BEC = 90^o - \frac{1}{2} \times \angle BAC.$
Proof:
$\angle BCE = \frac{1}{2} (180^o - 2\angle BCD ) = 90^o - \angle BCD$
Similarly, $\angle CBE = 90^o - \angle CBD$
Similar to the last theorem,
$\angle BCE + \angle BEC + \angle CBE = 180^o$
$90^o - \angle BCD + 90^o - \angle CBD + \angle BEC = 180^o$
$\angle BEC = \angle BCD + \angle CBD = 90^o - \frac{1}{2} \times \angle BAC. \blacksquare$
I noticed, if we add $\angle BDC$ and $\angle BEC$, we get 180, so that means the quadrilateral BDCE is inscribed in a circle.
Interestingly, this circle goes through one of the excenters, the incenter and two of the vertices of the triangle. (You can also see the other attached picture.
Midcircle.JPG
Midcircle.JPG (34.19KiB)Viewed 3245 times
Please let me know if this is useful in maths, or just another cool fact that is not that useful. Also, let me know if this is known in maths, or if I have found something new.

Thanks! :D
2+2 is 4, -1 that's 3 quick maths!

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