Midcircles!!!
Posted: Thu Apr 01, 2021 9:58 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 $E.$
1. Prove that, $\angle BDC = 90^o + \frac{1}{2} \times \angle BAC.$
Proof:
2. Prove that, $\angle BEC = 90^o - \frac{1}{2} \times \angle BAC.$
Proof:
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.
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!
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 (23.52KiB)Viewed 5680 times
Proof:
Proof:
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 (34.19KiB)Viewed 5680 times
Thanks!