Projections, an Altitude, and the Orthocenter

AB'

≈

6.13

B'C

≈

8.71

BB'

≈

14.35

HB'

≈

3.72

AB'

×

B'C

≈

53.39

BB'

×

HB'

≈

53.39

The intersection of the altitudes of a triangle is the orthocenter.

The product of the projections of two sides of a triangle onto the third side is equal to the product of the altitude to that side and the distance of the orthocenter to that side.

In terms of the figure, let H be the orthocenter of the triangle ABC and let B' be the intersection of BH and AC. Then the projections of AB and BC onto AC are AB' and B'C, the altitude is BB', and the distance from the orthocenter to AC is HB'. Then AB'×B'C = BB'×HB'.