##### Transversals in Triangles

Simple properties of medians, central verticals, angle bisectors and heights are shown.The idea is to show that it is not self-evident, that transversal lines in a triangle intersect in one point.

Author: Dieter Bergmann

Topic: Mathematics

Tags 2D shapes , Algebra , Analysis , Analytic Geometry , Animations , Calculus , Chaos , Derivative , Didactics , Dynamic Simulations , Dynamic Systems , Equations , Function Plotter , Functions , Geometry , Graphing , Sharing Inspiration 2017 , Solving equations , Triangle , Zeros

The idea is to show that it is not self-evident, that the corresponding transversal lines in a triangle intersect at one point. It is a special case that occurs only under certain conditions. This may not be new, but perhaps surprisingly. I read anyway nothing about it yet.

With the handheld this can be easily illustrated by simple trial and error. The vertices of the triangle as well as the point Â can be moved along one side of the triangle.

1 The medians (Problem 1)

They intersect at one point if and only if the transversal ends at the midpoint of the side. If the ratio is not 1:2, there is no common point of intersection. In the worksheet, you can move the point . Then the points on the other sides move so, that this relationship applies to all sides.

2 The central verticals (Problem 2)

They intersect at one point if and only if the vertical is at the midpoint of the side. If the ratio is not 1:2, there is no common point of intersection. In the worksheet, you can move the point. Then the points on the other sides move so, that this relationship applies to all sides.

3 The angle bisectors (Problem 3)

They intersect at one point if and only if the angles are devided in two equal parts. Is the ratio is not 1:2, there is no common point of intersection. In the worksheet, you can move the point . Then the points on the other sides move so, that this relationship applies to all angles.

4 The heights (Problem 4)

They intersect at one point if and only if the transversal und the side are perpendicular to one another. In the worksheet, you can move the point MoveMe. Then the points on the other sides move so, that the bias angle of the transversal und the side is the same.