Orthic triangle

(WIP)

In geometry, given any $\triangle ABC$ , let $D$ , $E$ , and $F$ denote the feet of the altitudes from $A$ , $B$ , and $C$ , respectively. Then, $\triangle DEF$ is called the orthic triangle of $\triangle ABC$ .

It's easy to see that there is no orthic triangle if $\triangle ABC$ is right. The only two cases are when $\triangle ABC$ is either acute or obtuse.

Orthic triangles are not unique to their mother triangles, as one acute and one to three obtuse triangles are guaranteed to have the same orthic triangle. To see this, take an acute triangle and swap its orthocenter and any vertex to get an obtuse triangle. It's easy to verify that the orthic triangle will remain the same. The number of obtuse triangles is determined by how many unique side lengths the given orthic triangle has: a scalene triangle has three, an isosceles triangle has two, and an equilateral has only one.

Cyclic quadrilaterals

In both the acute and obtuse case, quadrilaterals $ADEB$ , $BEFC$ , $CFDA$ , $AEHF$ , $BFHD$ , and $CDHE$ are cyclic.

Proof: we will be using directed angles, denoted by $\measuredangle$ instead of the conventional $\angle$ . We know that $\measuredangle ADB = 90^{\circ} = \measuredangle AEB,$ and thus $ADEB$ is cyclic. In addition, $\measuredangle AEH = \measuredangle AFH,$ so $AEHF$ is also cyclic. It follows that the other cyclic quadrilaterals are also cyclic. $\square$

These cyclic quadrilaterals show up frequently in olympiads and are the most crucial section of this article.

Incenter

If $\triangle ABC$ is acute, then the incenter of the orthic triangle is the orthocenter $H$ .

If $\triangle ABC$ is obtuse, then the incenter of the orthic triangle is the obtuse vertex.

Relationship with the incenter/excenter lemma

Lemma: In any $\triangle ABC$ , let $I_A$ denote the $A$ -excenter, and $I_B$ and $I_C$ defined similarly. Then $\triangle ABC$ is the orthic triangle of $\triangle I_A I_B I_C$ .

Proof:

This lemma also applies in reverse. In the acute case, $A$ , $B$ , and $C$ are the excenters of the orthic triangle, while in the obtuse case, the two vertexes with acute angles and the orthocenter of $\triangle ABC$ are the excenters.

With this in mind, we can transfer results about the incenter and excenters to the orthic triangle, in particular, the incenter/excenter lemma. Although, we must divide our investigation into two cases: acute and obtuse.

In the acute case, the incenter-excenter lemma tells that every cyclic quadrilateral of the orthic triangle has a circumcenter on the nine-point circle of $\triangle ABC$ . Quadrilaterals $AEHF$ , $BFHD$ , and $CDHE$ follow immediately from the lemma. As for $ADEB$ , $BEFC$ , and $CFDA$ , via the inscribed angle theorem, their circumcenters are the midpoints of the side lengths of $\triangle ABC$ , which we know to be on the nine-point circle.

In the obtuse case, the six cyclic quadrilaterals also have circumcenters on the nine-point circle of $\triangle ABC$ by similar logic.

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Orthic triangle

Contents

Cyclic quadrilaterals

Incenter

Relationship with the incenter/excenter lemma

See also