Isosceles Triangle
**ISOSCELES TRIANGLE**

*Balmoral Software*

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:

### Metrics

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).
### Incircle

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

### Inellipse

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,

### Circumellipse

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

### Circumcircle

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

**Figure** | **Parameters** | Perimeter | Area | Centroid |

Incircle | R = u(s - u)/v | 2πR | πR^{2} | (0,R) |

Inellipse | | Computed | | (0,v/3) |

Isosceles triangle | Width: 2u Height: v | 2(u + s) | uv | (0,v/3) |

Circumellipse | | Computed | | (0,v/3) |

Bounding circle | Obtuse: R = u | 2πu | πu^{2} | |

Acute: R = | 2πR | πR^{2} | (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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