This note gives the common calculations in the solution for problems titled Dynamic Geometry: P96,
so far by @Valentin Duringer. .
Radius, Chord Length, and Coordinates
Let the chord dividing a circle with center at into two circular segments of heights and be . Diameter through cut perpendicularly at . Then , , and diameter , therefore radius of the circle is . By intersecting chords theorem . If we set as the origin of the -plane with parallel to the -axis, then is along , and and .
Angles of Moving Vertices
Label the upper variable triangle as and the lower variable triangle . Since the inscribed angle and the central angle both form the same arc , . Similarly, .
Side Lengths of Triangle
Since the diameter of a circumcircle is given by the sine rule:
Similarly, , , and ,
The Largest Rectangle Inscribed by a Triangle
Consider any with a height and a width of . Let the rectangle inscribes be and the height of be , where . Then the area of rectangle , . Since and are similar . And . Then . By AM-GM inequality, , and equality occurs when . Then the rectangle has a maximum area , when the height and breadth of the rectangle are half of that of the triangle.
The Square Inscribed by a Triangle
Let us consider the relationship between the side length of a square to the height and width of the triangle that inscribes it. Note that and are similar. Let the height of be , then . From , we have
Similarly for the square in , .