Observations of Pedal Triangles
by
Hee Jung Kim
Let a triangle ABC and a point P is given. The pedal triangle for pedal point p is a triangle formed by connecting intersections of three sides of the triangle ABC with lines perpendicular to the sides of the triangle ABC and passing through the given point P. Let’s explore various pedal triangles for a given point P.
I. P is outside or inside of the triangle.
II. Here is a Script Tool for the general construction of a pedal triangle to triangle ABC, where P is an point.
III. The centroid of a triangle is an intersection of three medians.
Let P be the centroid of triangle ABC.
3.1 In general, no special features of the pedal triangle RST happens.
3.2. If triangle ABC is isosceles, so is the pedal triangle RST. In the figure below, we see that RS and AB are parallel.
IV. The incenter of a triangle is the point on the interior of the triangle that is equidistance from the three sides. The incenter is the center if incircle (the inscribed circle) the triangle.
Let P be the incenter of triangle ABC.
4.1. Since R, S, and T are on the line to each sides of triangle ABC respectively, m(PR)=m(PS)=m(PT) by definition of the incenter.
4.2. If triangle ABC is isosceles, so is the pedal triangle RST.
V. The orthocenter of a triangle is the intersection of three lies containing the altitudes.
Let P be the orthocenter of triangle ABC.
5.1. If triangle ABC is an acute triangle (all three angles are acute angles), then the pedal triangle RST is inside triangle ABC.
5.2. If triangle ABC is an acute isosceles triangle, so is the pedal triangle RST.
5.3. If triangle ABC is a right triangle, the pedal triangle RST is does not exist (or is collapsed into a line segment) because the orthocenter of a right triangle is on the vertex of the right angle.
5.4. If triangle ABC is an obtuse triangle (one of the angles is an obtuse angle), so the orthocenter is outside triangle ABC, then the pedal triangle RST is partially overlapped with triangle ABC.
5.5. If triangle ABC is an obtuse isosceles triangle, the pedal triangle RST is an acute isosceles triangle.
VI. The circumcenter of a triangle is the point equidistant from the three vertices of the triangle.
Let P be the circumcenter of triangle ABC.
6.1. If triangle ABC is an acute triangle, the pedal triangle RST is the medial triangle which is formed by connecting the three midpoints of the sides of the triangle ABC.
Since P is the circumcenter, it is the intersection of the three perpendicular bisectors of three sides. Therefore, the line perpendicular to each side and passing through P is in fact the perpendicular bisector of each side. That is why each point R, S, and T of the pedal triangle are the midpoints of three sides and consequently the pedal triangle is the medial triangle.
6.2. If triangle ABC is an obtuse triangle, so the circumcenter is outside triangle ABC, the pedal triangle RST is still the medial triangle of triangle ABC.
6.3. If triangle ABC is a right triangle with right angle A, then P is the midpoint of the hypotenuse BC, the opposite side of angle A. Therefore P is one of the vertices of the pedal triangle RSP, say S. The pedal triangle RST (or RPT) is also a right triangle.
VII. Let P be a point on a side, but not one of the vertices of triangle ABC. Then P is one of the vertices of the pedal triangle, say S.
7.1. If triangle ABC is an acute triangle, the pedal triangle RST is an obtuse triangle.
7.2. If triangle ABC is a right triangle or an obtuse triangle, the pedal triangle RST is collapsed into a line segment perpendicular to a side. Obviously, the same thing happens if P is one of the vertices of triangle ABC.
