Geometric Construction 2

Ilustrate this theorem: Let ABC be a given triangle and O an arbitrary point of the plane. Draw AO, BO, CO to meet BC, CA, AB in L, M, N respectively, and then draw MN, NL, LM to meet BC, CA, AB in U,V,W respectively. Then U,V,W are collinear.

The straight line passing through U,V,W is called the polar of the point O with respect to the triangle ABC.

Suppose the triangle ABC and a straight line in m the plane are given. Construct the unique point O having m as polar. This point is called the pole of the line m.

Illustrate this: Let ABCDE be an arbitrary pentagon, F the point of intersection of the nonadjacent sides AB and CD, M the point of intersection of the diagonal AD with the line EF. Then the point of interestion P with the side AE with the line BM, the point of intersection Q of the side DE with the line CM, and the point of intersection R of the side BC with the diagonal AD all lie on one line.

Illustrate Pappus Theorem: If the vertices of a hexagon fall alternatively on two lines, the intersections of opposite sides are collinear.

Illustrate Theorem of Desargues: If two triangles have a center of perspective, they have an axis of perspective.

Illustrate Theorem on Doubly Perspective Triangles: Two doubly perspective triangles are in fact triply perspective.

Illustrate Theorem on Triply Perspective Triangles: Two triply perspective triangles are in fact quadruply perspective.

Illustrate this theorem: Let O be a point on the plane of a triangle ABC, and let A₁, B₁, C₁ be the points of intersection of the lines AO, BO, CO with the sides of the triangle opposite A,B,C. If A₂, B₂, C₂ are on B₁C₁, C₁A₁, A₁B₁ such that the three lines A₁A₂, B₁B₂, C₁C₂ are concurrent, then AA₂, BB₂, CC₂ are concurrent.