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## High school geometry

### Course: High school geometry>Unit 8

Lesson 12: Constructing circumcircles & incircles

# Geometric constructions: triangle-inscribing circle

Sal constructs a circle that inscribes a given triangle using compass and straightedge. Created by Sal Khan.

## Want to join the conversation?

• The question asks to construct a circle inscribing the triangle.
So doesn't that mean the triangle should be inside the circle ?
The opposite goes for the next VDO : Constructing circumscribing circle
• It is an "inscribing" circle because the circle is inscribed inside the triangle. Agreed though, "Construct a circle inscribed in the triangle" would be more clear.
• The use of the 4 headed arrow is sub-optimal. Much better would be a hollow circle with short horizontal and vertical lines (about 1/2 radius) pointing inward so you could see exactly where you are placing the dot. Further: The drawing should be closer to the 'ideal' line and point.

This is important as multi-stage constructions have accumulated errors. E.g. in the video on an inscribed circle the speaker demos how to do it, and despite exercising reasonable care, his circle doesn't quite touch one side. This both introduces the idea that sloppy workmanship is tolerable, and introduces doubt as to the validity of the construction.

Also: There is no explanation as to why the construction is valid. Why is the intersection of two bisectors the centre of the circle? By what magic is this the right place for the third side.

The proof is not difficult, although it has many steps:

Bisect the angle.
Pick a point on the bisector.
From that point construct perpendiculars through that point to each of the two sides of the angle.

Show that the two triangles formed are congruent.

Since the point is arbitrary, it means that any point on the bisector is equidistant from both sides of the triangle.

Repeat for another angle.

Repeat the construction from the intersection to all sides. One of the perpendiculars will be a side of two different triangles. Equality is transitive so if A=B and B=C then A=C so all three lengths are equal.
• I agree that this does introduce the idea that "not quite", or "almost" is tolerable, but sometimes it is. The point here is to teach you how to do it, not to be a perfectionist. Khan academy usually teaches one method, the one they think is best. There are many different methods and proofs you can find online. If you can do a proof yourself, why does someone need to show you? Good question Sherwood.
• How does Sal know to take the two circles and place them like that? How does he know it will bisect that angle?
• Sal kind of skips the normal first step of starting at a vertex and drawing a circle, then drawing two congruent circles whose intersection would be equidistant from the two points. Sal avoids the first step by drawing two congruent circles whose radii are equidistant from the vertex. Does this help, or you need more explanation?
(1 vote)
• Why doesn't Sal use the perpendicular bisector method, by putting a compass on either vertex, to create the bisector?
• The point is to bisect the angle, no the line. Occasionally it does both at once, but not always. Although he is bisecting the line, that is not the goal.
(1 vote)
• What can j be doing wrong? I do all the steps (double checked ) and yet my circle is never the right size.
• what website is he using for these types of problems ? How could i access to these ? cuz i want to solve these questions myself first before i watch the video. Anyone got a link ? thankyou
• It is the old version of Khan Academy (I believe).
(1 vote)
• Can you construct a circle that is inscribed in a square?
(1 vote)
• Yes. Take a square with side length x. Draw the two diagonals. Put the compass at the intersection of the two diagonals and draw a circle with radius x/2. This will be the inscribed circle.