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Any line perpendicular to any chord of a circle and passing through its midpoint also passes through the circle's center. The butterfly theorem states that, if M is the midpoint of a chord PQ of a circle, through which two other chords AB and CD are drawn; AD and BC intersect chord PQ at X and Y correspondingly, then M is the midpoint of XY.
Let A 0, B 0, C 0 be the projections of A p, B p, C p onto BC, CA, AB, respectively. Then A 0, B 0, C 0 are collinear. Moreover, the new line passes through the midpoint of PH, where H is the orthocenter of ΔABC. If ℓ passes through P, the line coincides with the Simson line. [9] [10] [11] A projective version of a Simson line
Line DE bisects line AB at D, line EF is a perpendicular bisector of segment AD at C, and line EF is the interior bisector of right angle AED. In geometry, bisection is the division of something into two equal or congruent parts (having the same shape and size). Usually it involves a bisecting line, also called a bisector.
Given a line A and a point P on A, as well as two intersecting lines M and N, both parallel to A there exists a line G through P which intersects M but not N. Given a line A as well as two intersecting lines M and N, both parallel to A, there exists a line G which intersects A and M, but not N. Given a line A and two distinct intersecting lines ...
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A closed line segment includes both endpoints, while an open line segment excludes both endpoints; a half-open line segment includes exactly one of the endpoints. In geometry , a line segment is often denoted using an overline ( vinculum ) above the symbols for the two endpoints, such as in AB .
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The line perpendicular to the directrix and passing through the focus (that is, the line that splits the parabola through the middle) is called the "axis of symmetry". The point where the parabola intersects its axis of symmetry is called the "vertex" and is the point where the parabola is most sharply curved. The distance between the vertex ...