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If a quadrilateral has a perimeter, it is called a chordal quadrilateral - do your homework . All isosceles trapezoids, all rectangles and therefore all squares have a circumcircle.

The circumcircle of an n-corner is the circle that passes through all the vertices of the n-corner. The sides of the n-corner are chords of the circumcircle.

The following theorem applies to all chord quadrilaterals:

The sum of opposite angles in the chord quadrilateral is 180°.

The theorem is proved for the case that the centre of the circumcircle lies within the chord quadrilateral.

Prerequisite:

A, B, C and D lie on a circle around M, i.e. MA(average) =MB(average) =MC(average) =MD(average) =r

Assertion:

∢DAB+∢BCD=∢ABC+∢CDA=180°

Proof:

In the isosceles triangles - geometry problem solver - ABM, BCM, CDM and DAM the base angles are congruent to each other in pairs. Then ∢DAB+∢BCD=α+β+γ+δ= S and also ∢ABC+∢CDA=α+β+γ+δ= S.

Since the sum of interior angles in the quadrilateral is 360°, 2S=360°, so S=180°. (w. b. w.)

For the other two cases (M lies on one side of the chordal quadrilateral or outside the chordal quadrilateral) the proof is analogous - statistic homework help . All regular polygons have a circumcircle, which is often used to construct them. Polygons drawn in a perimeter are called inscribed polygons.

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