Assignment 8: Altitudes and Orthocenters

# Perform the following constructions

a. Construct any triangle ABC.

b. Construct the Orthocenter H of triangle ABC.

c. Construct the Orthocenter of triangle HBC.

d. Construct the Orthocenter of triangle HAB.

e. Construct the Orthocenter of triangle HAC.

f. Construct the Circumcircles of triangles ABC, HBC, HAB, and HAC.

Construct the nine point circles for triangles ABC, HBC, HAC, and HAB.

It appears that all the nine-points circles are the same circle. Let's see what happens when we display all of the circles at the same time...

Here I have changed the colors of the nine-points triangles for each triangle (pink is for HAB, purple is for HBC, lavender is for HAC, and orange is for ABC). For this exploration, I will show that triangle JLP is congruent to triangle NQK.

When constructing the nine-points triangle (for triangle ABC) points J, L, and P come from the midpoints of the sides of the triangles.  Then points I, M, and O come from the altitudes.

Points N, Q, and K are the midpoints of the altitudes. So triangle NQK is similar to triangle ABC (1:4).

Triangle QJP is similar to triangle HBC, so segment JP is proportional to BC (1:2). Similarly, segment JL is proportional to AC (1:2) and segment PL is proportional to AB (1:2). Since all the sides are proportional (by the same ratio) triangle JLP is also similar to triangle ABC.

So now we know that triangle JLP is similar to NQK (1:4). Since both triangles are similar to ABC and both have a ratio of their areas to the area of ABC of 1:4, the triangles JLP and NQK are congruent.

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