# Euclid's Muse

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# Search Results for “Proof”

By admin
##### Circle Proof
This is the first app ever on Euclid’s Muse! It provides a draggable diagram to help illustrate a mathematical proof. This proof was discovered when modeling the Twisted Savonius style wind turbine from a top view. The full proof can be found here.

Tags: Proof, circles, draggable, real-world, savonius

##### Proof by Shear Transformation
This app allows you to explore geometric proof that is a variation of the original Euclid’s proof. In this proof, the shear transformation is used to change the shapes without changing their areas.

Tags: Pythagorean-Theorem, shear-transformation, common-core, middle-school, geometry, proof

##### Garfield's Proof
This app allows you to explore the algebraic proof offered by James A. Garfield, the twentieth president of the United States. He discovered this proof five years before he became the President.

Tags: Pythagorean-Theorem, algebraic-proof, area, middle-school

##### Bhaskara’s Proof
This app allows you to explore algebraic proof by the ancient Hindu mathematician Bhaskara, who used the properties of similarity to prove the theorem.

Tags: Pythagorean-Theorem, algebraic-proof, right-triangle, middle-school, similarity, ratio

##### Pythagoras' Proof
This app allows you to explore geometric proof offered by Greek mathematician Pythagoras himself.

Tags: Pythagorean-Theorem, right-triangle, geometric-proof, middle-school

##### proof without words
A proof that the locus of the center of a circle tangential to two nested circles is an ellipse.

Tags: ellipse, focus

##### Fermat Toricelli Geometric Proof
We use rotations to construct a path equal in length to the sum of the distances AE+BE+CE, whose end points are fixed.  Minimum for AE+BE+CE occurs when this path is a straight line. So long as no angles of the original triangle are greater than 120 degrees, this works.  What goes wrong when an angle does exceed 120 degrees?

Tags: mechanics, fermat

##### Chord Angle Theorem
The chord angle theorem states that in an inscribed triangle (ABC) where A is the center of the circle and BC is a chord, and BDC is an inscribed triangle on the same chord, angle BDC must equal one half of angle BAC. Try changing the angle and moving point D and observe the theorem’s truth. Note: the measure of angle BDC is being constantly recalculated as point D is dragged, but it doesn't change because of this theorem.

Tags: Chord, Angle, Theorem, Circle, Draggable, Proof

##### Cyclic Quadrilateral Theorem
The Cyclic Quadrilateral Theorem states that for a quadrilateral inscribed in a circle, the measures of opposite angles must add to 180 degrees. Drag the points and observe the angle measures to see how this theorem holds true.

Tags: Quadrilateral, Circle, Cyclic-Quadrilateral, Theorem, Proof, Draggable

By admin
##### Pythagoras
This clock illustrates Euclid’s proof of the Pythagorean Theorem (as well as telling the time).  The hypotenuse of the triangle is the diameter aligned with the hour hand.  The vertex of the triangle lies on top of the minute hand.  Seconds are counted out by two triangles whose apexes move parallel to their bases (and thus preserve area).  Euclid's proof depends on observing that at the 30 second mark, these two triangles are congruent.

Tags: clock