Math 2.0

Geometry Definition

Incircle

Builds on IncentreCircle

Definition

Let ABCABC be a triangle in the Euclidean plane ℝ2\mathbb{R}^2, with incentre II and inradius rr. Let dd be the Euclidean distance. The incircle of ABCABC is the circle with the centre II and the radius rr: { X∈ℝ2  :  d(X,I)=r }.\{\, X \in \mathbb{R}^2 \;:\; d(X, I) = r \,\}.

Remarks

The incircle touches each side of the triangle at exactly one point, the foot of the perpendicular from II to that side. Let TT be the area of ABCABC and ss half of its perimeter. Then T=rsT = r s, so r=T/sr = T / s. An excircle is a circle outside the triangle that touches one side and the extensions of the other two sides. Each triangle has three excircles.

Reading path

To understand this page from the ground up, read these 3 pages in order. The foundations come first.

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Circle, concept, no evidence levelCircleconceptIncentre, concept, no evidence levelIncentreconceptIncircle, definition, no evidence levelIncircledefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Circle concept
  2. Incentre concept
  3. Incircle definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Incircle_and_excircles, lead section
  2. Web https://en.wikipedia.org/wiki/Incircle_and_excircles, Relation to area of the triangle
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