## Centers of Triangles

This is a two part exploration of question number eight of Dr. Wilson's assignment pages for EMAT 6680. Found by clicking here ->
The first part attempts to prove a triangle constructed from the midpoints of the altitudes of an acute triangle is similar to the parent triangle. In addition to proving that this triangle is similar to the parent triangle, another proof is used to show that the constructed triangle is congruent to the medial triangle.
The second part of the observation looks at the relationship of the circumcenter, orthocenter, incenter, and centroid of the parent triangle, and constructed triangle.
Click the links for the definitions of each term.

## Part I:

Given:
Triangle ABC
A' midpoint of HA
B' midpoint of HB
C' midpoint of HC

Prove triangle A'B'C' ~ triangle ABC

Proof:

By the definition of midpoint, HA'= A'A and HC'=C'C. Then HA = 2*HA' and AC = 2*HC'. Since the angle A'HC' and AHC are the same angle triangles HA'C' and HAC are similar by SAS Similarity Theorem. By the same method, this can be proven that triangles HB'C'~HBC and triangles HB'A'~HBA. From the first similarity relationship the ratio of A'C to AC is 1 to 2. This is also the case for the ratio of B'C' to BC and A'B' to AB. By the SSS similarity relationship, then you can conclude that triangle A'B'C'~ABC. Refer to the following figure for a visual and analytical demonstration of the proof.

Given:
Triangle ABC
D midpoint of AB
E midpoint of AC
F midpoint of BC

Triangle ABC ~ Triangle FDE

Proof:

With a similar procedure as the previous proof, we can use the definition of midpoint to create a relationship between sides AD, AB and AE, AC. By the definition of the midpoint, AD = 1/2 (AB) and AE = 1/2 (AC). Then the ratio of the two pairs of sides (AD/AB) and (AE/AC) are both 1 to 2 (1/2). Also since angle A is congruent to itself, we again have a SAS similarity relationship. Thus, triangle ADE ~ triangle ABC. By the same approach we can show that triangle CEF ~ triangle CAB and triangle BDF ~ triangle BAC. From these similar triangles the sides of the medial triangle (DE, EF, and FD) correspond to the larger triangle as well. (BC/DE) = (AB/EF) = (AC/DF) = (2/1). Therefore, triangle DEF ~ to triangle BCA.

Conclusion: By transitivity, since both triangles A'B'C', and DEF are similar to triangle ABC, implying that their corresponding angles are all congruent, and since the ratio of the sides are all 1 to 2 from triangles A'B'C', and DEF. It is enough to show that triangle DEF is congruent to triangle B'C'A'.

## Part II

After constructing triangle A'B'C' in the interior of triangle ABC, the incenter (I), circumcenter (C), centroid (G), and orthocenter (H), were constructed in both triangles.

1) Since triangle A'B'C' was constructed from the midpoints of the altitudes of triangle ABC, both triangles shared the same orthocenter.
2) The question only asked to take into consideration the situation for an acute triangle, upon observing the behavior of the I,C, G, and H the obtuse, and right triangle case the following could be said:
(GSP examples are given for each different case, click either obtuse, right, or acute to go to the link)
a. Due to the similarity of the triangles, the proportion of the segments composing of the orthocenter and incenter of each respective triangle will be in the same ratio as the ratio of sides for each triangle.
b. Due to the similarity of the triangles, the proportion of the segments composing of the orthocenter and centroid of each respective triangle will be in the same ratio as the ratio of sides for each triangle.
c. Due to the similarity of the triangles, the proportion of the segments composing of the orthocenter and circumcenter of each respective triangle will be in the same ratio as the ratio of sides for each triangle.
d. In an obtuse triangle condition, triangle A'B'C' will be found exterior to triangle ABC.
e. In the right triangle condition, triangle A'B'C' there is one case where the orthocenter of the triangles will be at the right angle.
f. In the acute triangle condition, triangle A'B'C' will be found in the interior of triangle ABC.
3) Upon constructing additional similar triangles much of the properties will remain the same (i.e. similar ratio of sides and the segments of AB/A'B', HI/HI', HG/HG', and HC/HC'). Click here ->For a GSP interpretation.

Obtuse Triangle Case:

Acute Triangle Case:

Right triangle Case: