Altitudes and Orthocenter

By Na Young Kwon

11. Construct any acute triangle ABC and its circumcircle.

Construct the three altitudes AD, BE, and CF. Extend each altitude

to its intersection with the circumcircle at corresponding points

P, Q, and R.

Picture

Find .___(1)

We can know the result is 4 by calculating GSP.

If a triangle is a regular triangle, we can easily know the result

of is 4. Because in a regular triangle the circumcenter

is congruent to the orthocenter and a distance from a vertex to a

circumcenter is two times of a distance from a circumcenter to

an opposite side.

What is the value of in acute triangle ABC?

Let’s prove this result.

We can find the result from the pedal triangle DEF and a triangle PQR.

Length of HD=Length of DP

Because a triangle DEF is similar to a triangle PQR and the ratio is 1:2.

So we change the equation (1) the following.

We can get the following result about the area of a triangle ABC.

Area of ABC= BC * AD * 1/2 ---(1)

= AC * BE * 1/2 ----(2)

= AB * CF * 1/2 ----(3)

And area of ABC = area of HBC + area of HCA + area of HAB

= BC*HD*1/2+AC*HE*1/2+AB*HF*1/2 – (4)

BC * AD = BC * HD+AC * HE+AB * HF

By (1),(2) and (3)

So we can know the result of equation (1)

= 4

We can apply this result to the obuse triangle.

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