Assignment 8: Altitudes and Orthocenter

For this exploration we will extend the altitudes of ΔABC to meet the circumcircle at the points P, Q, and R. Then we will find the relationship between ΔPQR and the orthic triangle ΔDEF.

To find and prove the relationship between the two triangles let's first look at the relationship between the distance from the orthocenter to the foot of the altitude and the distance from the foot of the altitude and the corresponding point on the circumcircle. If we were able to make proofs by just eyeballing it we would be able to skip a bunch of steps and say they are equal in distance, but that would not be very much fun so let's prove it with what we know about circles and angles. We see that the two angles ∠ARC and ∠ABC are congruent because they are two angles inside the circumcircle cutting the arc AC. Now take a look at the model below and to the left. We can see that ΔABC is being intersected by two segments at right angles (the altitudes from the vertices A and C). Because these two segments, CF and AD, intersect BC and AB at right angles the two segments form two pairs of congruent angles. Furthermore, the angles, ∠AFO and ∠COD (where O is the orthocenter) are not only congruent to each other but congruent to ∠ABC which is equal to ∠ARC. To summarize: ∠AFO ≅ ∠COD ≅ ∠ABC ≅ ∠ARC. Therefore, we can say that ΔARO is an isosceles triangle with AF as a segment bisector hence OF = RF. A similar proof is used to show that OE = EQ and OD = DP. We now can conclude that the distance from the orthocenter to a vertex of the orthic triangle is one half the distance from the orthocenter to a corresponding vertex of the extended altitudes. Therefore, the relationship between ΔDEF and ΔPQR is 1:2.

Below, we can see that the relationship does not hold when ΔABC is an obtuse triangle. For from the proof of minimum perimeter of a triangle inscribed in triangle shows us that the orthic triangle is the triangle with this minimum perimeter if and only if the triangle is acute. The orthic triangle ceases to exists when ΔABC is no longer acute. Although, ΔPQR may still exist the relationship, 1:2, between the two triangles can no longer exist. Please reference my Assignment 6 to see this proof of minimum perimeter.