Khan Academy on a Stick
Special properties and parts of triangles
You probably like triangles. You think they are useful. They show up a lot. What you'll see in this topic is that they are far more magical and mystical than you ever imagined!
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Circumcenter of a Triangle
ccMultiple proofs showing that a point is on a perpendicular bisector of a segment if and only if it is equidistant from the endpoints. Using this to establish the circumcenter, circumradius, and circumcircle for a triangle
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Circumcenter of a Right Triangle
ccShowing that the midpoint of the hypotenuse is the circumcenter
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Three Points Defining a Circle
ccThree points uniquely define a circle. The center of a circle is the circumcenter of any triangle the circle is circumscribed about.
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Area Circumradius Formula Proof
ccProof of the formula relating the area of a triangle to its circumradius
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2003 AIME II Problem 7
ccArea of rhombus from circumradii of triangles
Perpendicular bisectors
In this tutorial, we study lines that are perpendicular to the sides of a triangle and divide them in two (perpendicular bisectors). As we'll prove, they intersect at a unique point called the cicumcenter (which, quite amazingly, is equidistant to the vertices). We can then create a circle (circumcircle) centered at this point that goes through all the vertices. This tutorial is the extension of the core narrative of the Geometry "course". After this, you might want to look at the tutorial on angle bisectors.
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Point Line Distance and Angle Bisectors
ccThinking about the distance between a point and a line. Proof that a point on an angle bisector is equidistant to the sides of the angle and a point equidistant to the sides is on an angle bisector
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Incenter and incircles of a triangle
ccUsing angle bisectors to find the incenter and incircle of a triangle
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Inradius Perimeter and Area
ccShowing that area is equal to inradius times semiperimeter
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Angle Bisector Theorem Proof
ccWhat the angle bisector theorem is and its proof
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Angle Bisector Theorem Examples
ccUsing the angle bisector theorem to solve for sides of a triangle
Angle bisectors
This tutorial experiments with lines that divide the angles of a triangle in two (angle bisectors). As we'll prove, all three angle bisectors actually intersect at one point called the incenter (amazing!). We'll also prove that this incenter is equidistant from the sides of the triangle (even more amazing!). This allows us to create a circle centered at the incenter that is tangent to the sides of the triangle (not surprisingly called the "incircle").
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Triangle Medians and Centroids
ccSeeing that the centroid is 2/3 of the way along every median
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Triangle Medians and Centroids (2D Proof)
ccShowing that the centroid is 2/3 of the way along a median
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Medians divide into smaller triangles of equal area
ccShowing that the three medians of a triangle divide it into six smaller triangles of equal area. Brief discussion of the centroid as well
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Exploring Medial Triangles
ccWhat a medial triangle is and its properties
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Proving that the Centroid is 2-3rds along the Median
ccShowing that the centroid divides each median into segments with a 2:1 ratio (or that the centroid is 2/3 along the median)
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Median Centroid Right Triangle Example
ccExample involving properties of medians
Medians and centroids
You've explored perpendicular bisectors and angle bisectors, but you're craving to study lines that intersect the vertices of a a triangle AND bisect the opposite sides. Well, you're luck because that (medians) is what we are going to study in this tutorial. We'll prove here that the medians intersect at a unique point (amazing!) called the centroid and divide the triangle into six mini triangles of equal area (even more amazing!). The centroid also always happens to divide all the medians in segments with lengths at a 1:2 ration (stupendous!).
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Proof - Triangle Altitudes are Concurrent (Orthocenter)
ccShowing that any triangle can be the medial triangle for some larger triangle. Using this to show that the altitudes of a triangle are concurrent (at the orthocenter).
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Common Orthocenter and Centroid
ccShowing that a triangle with the same point as the orthocenter and centroid is equilateral
Altitudes
Ok. You knew triangles where cool, but you never imagined they were this cool! Well, this tutorial will take things even further. After perpendicular bisectors, angle bisector and medians, the only other thing (that I can think of) is a line that intersects a vertex and the opposite side (called an altitude). As we'll see, these are just as cool as the rest and, as you may have guessed, intersect at a unique point called the orthocenter (unbelievable!).
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Review of Triangle Properties
ccComparing perpendicular bisectors to angle bisectors to medians to altitudes
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Euler Line
ccThe magic and mystery of the Euler Line
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Euler's Line Proof
ccProving the somewhat mystical result that the circumcenter, centroid and orthocenter
Bringing it all together
This tutorial brings together all of the major ideas in this topic. First, it starts off with a light-weight review of the various ideas in the topic. It then goes into a heavy-weight proof of a truly, truly, truly amazing idea. It was amazing enough that orthocenters, circumcenters, and centroids exist , but we'll see in the videos on Euler lines that they sit on the same line themselves (incenters must be feeling lonely)!!!!!!!