section_path: Related figures/Figures inscribed in a triangle
section_name: Figures inscribed in a triangle
paragraph_text: Figures inscribed in a triangle
  
  As discussed above, every triangle has a unique inscribed circle (incircle) that is interior to the triangle and tangent to all three sides. Every triangle has a unique Steiner inellipse which is interior to the triangle and tangent at the midpoints of the sides. Marden's theorem shows how to find the foci of this ellipse. This ellipse has the greatest area of any ellipse tangent to all three sides of the triangle. The Mandart inellipse of a triangle is the ellipse inscribed within the triangle tangent to its sides at the contact points of its excircles. For any ellipse inscribed in a triangle A B C {\displaystyle ABC}, let the foci be P {\displaystyle P} and Q {\displaystyle Q}, then: P A ¯ ⋅ Q A ¯ C A ¯ ⋅ A B ¯ + P B ¯ ⋅ Q B ¯ A B ¯ ⋅ B C ¯ + P C ¯ ⋅ Q C ¯ B C ¯ ⋅ C A ¯ = 1. {\displaystyle {\frac {{\overline {PA}}\cdot {\overline {QA}}}{{\overline {CA}}\cdot {\overline {AB}}}}+{\frac {{\overline {PB}}\cdot {\overline {QB}}}{{\overline {AB}}\cdot {\overline {BC}}}}+{\frac {{\overline {PC}}\cdot {\overline {QC}}}{{\overline {BC}}\cdot {\overline {CA}}}}=1.}
  
  From an interior point in a reference triangle, the nearest points on the three sides serve as the vertices of the pedal triangle of that point. If the interior point is the circumcenter of the reference triangle, the vertices of the pedal triangle are the midpoints of the reference triangle's sides, and so the pedal triangle is called the midpoint triangle or medial triangle. The midpoint triangle subdivides the reference triangle into four congruent triangles which are similar to the reference triangle.
  
  The intouch triangle of a reference triangle has its vertices at the three points of tangency of the reference triangle's sides with its incircle. The extouch triangle of a reference triangle has its vertices at the points of tangency of the reference triangle's excircles with its sides (not extended).
  
  Every acute triangle has three inscribed squares (squares in its interior such that all four of a square's vertices lie on a side of the triangle, so two of them lie on the same side and hence one side of the square coincides with part of a side of the triangle). In a right triangle two of the squares coincide and have a vertex at the triangle's right angle, so a right triangle has only two distinct inscribed squares. An obtuse triangle has only one inscribed square, with a side coinciding with part of the triangle's longest side. Within a given triangle, a longer common side is associated with a smaller inscribed square. If an inscribed square has a side of length q a {\displaystyle q_{a}} and the triangle has a side of length a {\displaystyle a}, part of which side coincides with a side of the square, then q a {\displaystyle q_{a}}, a {\displaystyle a}, h a {\displaystyle h_{a}} from the side a {\displaystyle a}, and the triangle's area T {\displaystyle T} are related according to q a = 2 T a a 2 + 2 T = a h a a + h a. {\displaystyle q_{a}={\frac {2Ta}{a^{2}+2T}}={\frac {ah_{a}}{a+h_{a}}}.} The largest possible ratio of the area of the inscribed square to the area of the triangle is 1/2, which occurs when a 2 = 2 T {\displaystyle a^{2}=2T}, q = a / 2 {\displaystyle q=a/2}, and the altitude of the triangle from the base of length a {\displaystyle a} is equal to a {\displaystyle a}. The smallest possible ratio of the side of one inscribed square to the side of another in the same non-obtuse triangle is 2 2 / 3 {\displaystyle 2{\sqrt {2}}/3}. Both of these extreme cases occur for the isosceles right triangle.
  
  The Lemoine hexagon is a cyclic hexagon with vertices given by the six intersections of the sides of a triangle with the three lines that are parallel to the sides and that pass through its symmedian point. In either its simple form or its self-intersecting form, the Lemoine hexagon is interior to the triangle with two vertices on each side of the triangle.
  
  Every convex polygon with area T {\displaystyle T} can be inscribed in a triangle of area at most equal to 2 T {\displaystyle 2T}. Equality holds only if the polygon is a parallelogram.