# Euclid's Muse

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# Search Results for “intersections”

##### Tridecagon Diagonals, Circles and Tangents
You'll want to start out with the heptagon and work your way up. This one's the same as all the others, just with a 13-sided regular polygon. Observe the tangencies to diagonals of circles centered at intersections of diagonals, when the circles are resized (by dragging). This is a smaller version that works well on most monitors (zoom in with two-finger touch). Bigger version here.

Tags: Tridecagon, diagonals, circles, tangents, intersections, puzzler, intricate, confusing, wow

By Duncan
##### circle intersections
Can you conjecture a formula for the product of the two distances from a point to a circle?

Tags: circle, tangent

##### A geometry theorem
H1,H2 and H3 are feet of altitudes, M1,M2,M3 are midpoints. H is the orthocenter.  X2,X3,Y1,Y3,Z1,Z2 are formed by reflecting H1,H2 and H3 in the other altitudes.  X,Y,Z are the intersections between the lines joining midpoints to these reflected points. Observe that H appears to be the incenter of XYZ so long as ABC is acute angled. Otherwise it appears to be one of the excenters

Tags: orthocenter, incenter

##### Circle Pedal
Given a curve and a pole point, the pedal is the locus of the intersections between the tangents to the curve and the perpendiculars through the pole point.  In this case the given curve is a circle, and the pole is the red point.  Try dragging the pole and observe how the clock face changes.  Press the Explain button to see the situation more clearly.

Tags: clock

##### Polar Line
The locus of the intersections of the tangents at the end of chords of an ellipse which pass through a common point is a straight line. A bit of a mouthful, but play with the diagram and it will become clear.

Tags: locus, polar, ellipse, conic

##### Hyperbola Polar Line
The polar line is the locus of the intersections of tangent lines at the ends of chords of teh parabola through a fixed point. Turns out to be conceptually important - not just a curiosity.

Tags: polar, conic, hyperbola