How Well Do You Know Your Geometry?

Geometry Level 2

How many triangles $ABC$ are there with integer side lengths such that the area of the triangle formed by joining the orthocenter, the circumcenter and the centroid of $\triangle ABC$ is $44$ square units?

Details and assumptions:

The orthocenter of $ABC$ is the point at which the altitudes of $ABC$ intersect.

The circumcenter of $ABC$ is the point which is equidistant from $A$ , $B$ and $C$ .

The centroid of $ABC$ is the point at which the medians of $ABC$ intersect.

The answer is 0.

2 solutions

Aryan Goyat
Jul 30, 2015

Orthocenter, centroid and circumcenter always lie on same line (except in case of equilateral triangle), hence formation of triangle is not possible.

O---------2x------G-----x---C

where

O represents orthocenter.

G represents centroid .

And C represents circumcenter.

Trevor Arashiro
Aug 13, 2014

Ok, this problem really made me take a second look at my common sense level. I prepared myself for a lot of work by finding the distances between the three points, but then I realized that the distance between the circumcenter to the centroid + the distance from the orthocenter to the centroid= the distance from the circumcenter to the orthocenter , meaning that the triangle would be degenerate and have area 0. Then I got the problem correct, looked at the tags, and couldn't help but laugh. Especially since I had already known what the Euler line was, I just never thought of it.

Same happened to me

Charuka Bandara - 5 years, 3 months ago

Very easy question Of logical sense

Manoj Gupta - 2 years, 4 months ago

How Well Do You Know Your Geometry?

The answer is 0.

2 solutions

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