Isosceles Triangle

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Consider a y-symmetric isosceles triangle with vertices (0,v) and (±u,0), u,v > 0. The width x height of its bounding rectangle is 2u x v.
Let t represent the usual parameter, ranging from 0 to 2π through the quadrants in counterclockwise order. The parametric equations of the triangle are piecewise linear in t:


The lateral side length of the triangle is
so its perimeter is 2(u + s) and its area is uv. Its centroid is the average of its vertex coordinates: (0,v/3).


The radius and center ordinate of the triangle's largest bounded circle (its incircle) are
R = d = u(s - u)/v


Using z = 0 in Lemma E,
d/dt x(t)[y(t) - z] = d/dt uv(1 - 2t/π)(1 - |1 - 2t/π|)
has a zero over [0,π) at t* = π/4. The corresponding coordinates are:
x* = u/2

y* = v/2,

so the inellipse dimensions are
For verification,


Using h = v and w = 2u in Lemma TE, the parameters of the circumellipse are


If the triangle is obtuse (u > v), its apex point (0,v) is not tangent to its bounding circle. In that case, the triangle's smallest bounding circle has a diameter between the vertices (±u,0) and its center at the origin.

If the triangle is acute (u ≤ v), then using h = v and w = 2u in Lemma TC, the circumellipse has radius and center ordinate

Summary Table

IncircleR = u(s - u)/v2πRπR2(0,R)
Isosceles triangleWidth: 2u
Height: v
2(u + s)uv(0,v/3)
Bounding circleObtuse: R = u2πuπu2
Acute: R = 2πRπR2(0,v - R)
The centers of the boundary ellipses are both the same as the centroid of the isosceles triangle, and the circumellipse is exactly twice as large as the inellipse.

Specific Example

The boundary conics for an obtuse isosceles triangle with u = 4 and v = 3 are shown below:

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