Perpendicular Bisector Theorem Explained (Proof) #circletheorem #mathematics #gcse #geometry

Here we provide proof that the perpendicular bisector of any chord in a circle, must pass through the circle center. The proof is based on the fact that, in an isosceles triangle, the perpendicular bisector of the base passes through the triangle vertext. If one radius is drawn from each end of the chord to the circle center, the result is an isosceles triangle where the sides are the radius of the triangle, and the vertex opposite the base, which is in the same position as the circle center. By bisecing the base of that triangle with a perpendicular line, the line must pass through the triangle vertex opposite the base, which happens to be the circle centerposition. The base of the triangle IS the chord, so the perpendicular bisects the chord. The perpendicular passes through the triangle vertext opposite the base, and that vertex is at circle center, so the perpendicular bisector of the chord passes through circle center. No other line perpendicular to the base of the triangle passes through the triangle vertex, so no other line is a perpendicular bisector. And since there is only one position for the perpendicular bisector, and at that position the bisector passes through the triangle vertex, which is also at the circle center, the perpendicular bisector of the base must also pass through the circle center. And given that the base of the triangle is also the chord, then the perpendicular bisector of the chord must pass through the circle center. Two video related to this subject: Perpendicular Bisector of The Base Passes Through The Triangle Vertex - EXPLAINED! https://youtu.be/Su6sudFUQ50 A different way to prove the theorem: https://youtu.be/Ne0l6VDCxow Chapters: - 0:00 intro 1:16 creating the proof structure - an isosceles triangle 2:20 adding and explaining the perpendicular bisector of the triangle base 2:48 the perpendicular bisects the chord and passes through circle center 3:13 do any non bisecting perpendicular lines pass through the circle center? Subscribe to TinyMaths here: https://www.youtube.com/c/TinyMaths/videos?sub_confirmation=1