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Symbolic computations on this page use Nerdamer Symbolic JavaScript to complement the in-built CAS of GXWeb

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GXWeb inCircles Explorer

Circles inside Triangles A Very Special inCircle GXWeb inCircles Explorer Mathematical Toolkit Build your own Model with GXWeb Every triangle has an incircle tangent to each side. This may be constructed by taking the perpendicular bisectors of each vertex to find the center, and the perpendicular from this center to each side for the radii. But how might we find the radius and area of such circles? GXWeb algebraic geometry software makes it easy to construct and find both numeric and algebraic solutions to such a problem, as shown in the YouTube videos below.

Jump to step-by-step instructions to easily create your own inCircles with GXWeb!

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Circles inside Triangles

Let \(r = DE = DF = DG \) be the radius of the circle in the model shown here. Then side AB is divided into segments AF \(( = a-r)\) and BF \(( = r)\), and side BC is divided into BG \(( = r)\) and GC \(( = b-r)\). But this also means that \(AE = a-r\) and \(CE = b-r\)! (Why?) And so \(AC = c = a + b - 2\cdot r\). \[c = \sqrt{(a^2 + b^2)} = a + b - 2\cdot r\] By this method... \[r = \frac{a+b-c}{2}\] \[r = \frac{(a+b-c)\cdot (a+b+c)}{2\cdot (a+b+c)}\]\[ = \frac{(a^2 + b^2 - c^2 + 2\cdot a\cdot b)}{2\cdot (a+b+c)}\] But \(a^2 + b^2 = c^2\), so... \[ r = \frac{2\cdot a\cdot b}{2\cdot (a+b+c)}\] Thus another form for the radius of the incircle of any right-angled triangle: 2 times the area of the triangle divided by the sum of the three sides! \[r = \frac{2\cdot (\frac{1}{2}\cdot a\cdot b)}{a + b + \sqrt{(a^2+b^2)}} \]\[= \frac{a\cdot b}{a+b+c}\]

For any triangle, the radius \(r\) is found by

\[r = \sqrt{\frac{(a+b-c)\cdot (a-b+c)\cdot (b+c-a)}{4\cdot (a+b+c)}}\]

The area of the incircle follows:

\[Area = \pi \cdot r^2 \]\[= \frac{\pi}{4}\cdot (a+b-c)^2 = \pi\cdot (\frac{a\cdot b}{a+b+c})^2 \ (for \ right \ angled \ triangles) \] \[= \pi \cdot (\frac{(a+b-c)\cdot (a-b+c)\cdot (b+c-a)}{4\cdot (a+b+c)}) \ (for \ any \ triangle)\]

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inCircles Explorer

Mathematical Toolkit

Build your own Model with GXWeb

 

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A Very Special inCircle

One triangle in particular has a special incircle: the simple 3-4-5 right-angled triangle has an incircle with radius exactly 1, and hence an area of exactly \(\pi\)!

Use the GXWeb inCircle Explorer or GXWeb to see if you can find any other triangles with a unit incircle radius, both right angled and general.

Begin with the first few Pythagorean triples, and see what you notice.

\[a = 3, b = 4, c = 5\] \[a = 5, b = 12, c = 13\] \[a = 7, b = 24, c = 25\] \[a = 8, b = 15, c = 17\] \[a = 20, b = 21, c = 29\]

Next, try these: they are all close, but are any exact?

\[a = 5; b = \frac{8}{3}; c = \sqrt{34}\] \[a = \sqrt{6}; b = 6; c = \sqrt{45}\] \[a = 7; b = \frac{22}{9}; c = \sqrt{53}\] \[a = b = c = \sqrt{12}\] \[a = 4, b = 7, c = 10\]

 

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inCircles Explorer

Mathematical Toolkit

Build your own Model with GXWeb

 

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GXWeb inCircles Explorer

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inCircles Explorer

Mathematical Toolkit

Build your own Model with GXWeb

Show General inCircle Formulae ⇑

This document requires an HTML5-compliant browser.	Right Angled General
\[AB = a\] ▶ 0 0 30 \[BC = b\] ▶ 0 0 30 \[AC = c\] ▶ 0 0 30	RESET Test Yourself

App generated by Geometry Expressions

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inCircles Explorer

Mathematical Toolkit

Build your own Model with GXWeb

 

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Mathematical ToolKit

About the MathBoxes...

Hint: When entering mathematical expressions in the math boxes below (f(x), g(x) and h(x)), use the space key to step out of fractions, powers, etc. On Android, begin entry by pressing Enter.

Type simple mathematical expressions and equations as you would normally enter these: for example, "x^2[space]-4x+3", and "2/3[space]". For more interesting elements, use Latex notation (prefix commands such as "sqrt" and "nthroot3" with a backslash (\\)): for example: "\\sqrt(2)[space][space]". Learn More?

 

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inCircles Explorer

Mathematical Toolkit

 

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Construct your own Model with GXWeb

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inCircles Explorer

Mathematical Toolkit

Try the Full GXWeb inCircle Construction

 

GXWeb inCircle Construction 2: Right Angled Triangle

GXWeb inCircle Construction 2: General Triangle

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inCircles Explorer

Mathematical Toolkit

 

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©2024 Compass Learning Technologies ← GXWeb Showcase → GXWeb inCircles Explorer

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