Petal Triangles

By Taylor Adams

A petal triangle of a given triangle
ABC is formed by perpendiculars of the sides of triangle ABC through a given
point P.

What happens if P is the centroid of
the triangle?

Because the centroid of the circle is
constructed by connecting the medians of the sides to the opposite vertices, it
is always in the middle of the triangle.
If P lies on the centroid of the triangle, the petal triangle will
always be on the inside of the triangle.
Click here to further explore what happens when P is on the centroid of
the circle.

What happens to the petal triangle
when P is on the incenter?

The incenter of a triangle is made by
intersection of the angle bisectors. A
circle can be constructed by the intersection of the angle bisectors with the
sides of the triangle. This is the
incircle of the triangle, and the incenter is the center of this circle.

We can see from the picture above
that the vertices of the petal triangle, when P is on the incenter of the
circle, lie on the incircle. Therefore,
the vertices of the petal triangle are equidistant from the incenter, or point
P. Click here to further explore what
happens when P is on the incenter of the circle.

What happens when P is Orthocenter?

The orthocenter of a triangle is
formed from the intersection of perpendiculars, or altitudes, of each
side. Thus, the orthocenter can fall on
the inside or outside of the original triangle (in red). In either case, the petal triangle (in pink) formed
when P is on the orthocenter is also the orthic triangle. The orthic triangle is formed by the
intersection of the altitudes with the sides of the original triangle. This is easy to see when the orthocenter is
on the inside of the triangle, like in the first picture. When the orthocenter is on the outside of the
triangle, the orthic triangle’s vertices lie on the extension of the sides of
the original triangle. Click here to
further explore what happens when P is on the orthocenter of the circle.

What happens when P is the
Circumcenter?

The circumcenter is formed by the intersection
of the perpendicular bisectors of each side of the triangle. Therefore, if P lies on the circumcenter of
the triangle, the petal triangle becomes the medial triangle. The medial triangle is formed by the
midpoints of a triangle. Since the circumcenter
is formed by the perpendicular bisectors, the perpendiculars that would form the
petal triangle are also the midpoints of the sides. Click here to further explore what happens
when P is on the circumcenter of the circle.

What happens when P is the center of
the Nine Point Circle of the original triangle?

With a general triangle, there is
nothing particularly interesting about the petal triangle formed when P is the
center of the nine point circle.

However, when the original triangle
is equilateral,

the petal triangle falls on the nine
point circle.

Click here to further explore what
happens when P is on the center of the nine point circle.