One of the most important properties about excircles

In this video, we present a solution to Iran TST 2015/2.
00:00 Problem Statement
00:41 Solution
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Solution PDF: calimath.org/p...
branch: geometry
difficulty: 7
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[@user-kj6lj7dp9j]

wonderful video !

[@calimath6701]

Thanks!

[@migorpresents8991]

So, in other words you're saying that triangles TNC and TNC' are equal, which further implies that TC = TC'. But how are these traingles equal? We only have a common side TN, NC = NC' and

[@calimath6701]

It is only enough if the angle is obtuse. This is given in our case, since N and C lie on different sides of SI_B and

[@Cooososoo]

How do you construct diagrams so perfectly

[@calimath6701]

Just a large compass and ruler and many attempts :D

[@mohammedmarzuqrahman8605]

Why TB × TC = TS^2 ? 3:32

[@calimath6701]

At this point, we haven't proven it; it's just a guess used to motivate the upcoming part of the video. The reason behind the guess is that both I_B and S are intersections of TM with the circle, indicating they should play similar roles. Therefore, if TB × TC = TB_I^2 holds true, then we anticipate TB × TC = TS^2 to be true as well.

[@Mather4269mixti]

You can also prove this using Ib as a circle of radius 0😅