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.

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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 triangleís 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.

 

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