Lester circle: Construction of the points

Construction of the points

In each of the applets below, the original triangle is gray and its vertices are red. Drag the vertices to experiment with triangles of different shapes. If you drag a point off the applet, you can reset the applet by pressing r on your keyboard.

Circumcentre C

Construct the perpendicular bisectors of the sides of the triangle; they meet at the circumcentre.

The circumcentre is the centre of the circumcircle, the circle through the three vertices.

(This construction can be used to draw the circle through any three points not on a line: just draw the circumcircle of the triangle they determine.)

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Nine-point centre N

Construct the altitudes of the triangle; they meet at the orthocentre. Find the following nine points:
- the feet of the altitudes •
- the points midway between the orthocentre and each of the vertices •
- the midpoints of the sides •.
These points all lie on the nine-point circle; draw it by constructing the circle through any three of the nine points. Its centre is the nine-point centre.

First Fermat point F

Construct equilateral triangles on the outside sides of the triangle. Join each vertex of the original triangle to the apex of the equilateral triangle on the opposite side. The three joining lines meet at the first Fermat point.

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Second Fermat point S

Construct equilateral triangles on the inside sides of the triangle. Join each vertex of the original triangle to the apex of the equilateral triangle on the opposite side. The three joining lines meet at the second Fermat point.

A word of caution about the Fermat points: the equilateral triangle on each side of the original triangle is on the outside if it’s on the opposite side of the triangle side from the original triangle, and on the inside if it’s on the same side of the triangle side as the original triangle. When constructing these triangles with dynamic geometry software, be careful that the two don't change places when a vertex of the original triangle is dragged through the opposite side. The trick is to link the construction of the equilateral triangles to the sign of the angles of the original triangle.

Lester circle:

Constructions