[size=150][left][br][b][size=200]The Triangle Midsegment Theorem[br][/size][/b][br][size=200]In a triangle, the segment joining the midpoints of any two sides will be parallel to the third side and half its length.”[br]Model in GeoGebra and prove![/size][/left][/size]

[size=200][b]Varignon’s theorem[/b] [br]“The midpoints of the sides of an arbitrary quadrilateral form a parallelogram.” Model in GeoGebra and prove![/size]

[size=200][b]Five concyclic points of a right angle triangle[/b] (The Circle Theorem of Apollonius) [br][br]ABC is a right angle triangle with the right angle at A. Prove that all the following points lie on the common circle: the vertex A, foot A0 of the altitude from A and centres Sa, Sb a Sc of the sides of the triangle ABC, see figure. [/size]

[size=200][b]A common point of three chords.[/b] [br]Suppose we have three circles in the plane, each intersecting the other two twice, but with no point common to all three, see figure. The three chords determined by these circles meet in a point. Prove![/size]

[size=200][b]The Right Angle Bisector Theorem [br][/b]“The internal bisector of the right angle of a right triangle bisects the square on the hypotenuse.” Model in GeoGebra and prove![/size]

[size=200]The squares ABCD and BFGE has the vertex B in common, see figure. Prove that segments AE and CF are perpendicular for each such two squares. [/size]

[size=200]Prove that for any parallelogram ABCD with sides a, b and diagonals e, f, see figure, the following equality holds true. [/size]