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The Encyclopedia of Geometry (0217)
Problem A quadrilateral formed by connecting the points where two straight lines that pass through the intersection of the diagonals of a parallelogram and intersect with each other at right angles intersect with each side is a rhombus. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$…
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The Encyclopedia of Geometry (0216)
Problem If the intersection of the bisectors of the two vertex angles $B$ and $C$ of the rectangle $ABCD$ is $E$, and similarly, the intersections of the bisectors of $A$ and $B$, $D$ and $A$, and $C$ and $D$ are $F, \ G$, and $H$, respectively, then the quadrilateral $FEHG$ is a square. $$ $$ $$ $$…
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The Encyclopedia of Geometry (0215)
Problem A convex quadrilateral $ABCD$ is a rhombus if $∠A=∠C, \ ∠B=∠D$ and $AB=BC$. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution The sum of the interior angles of a quadrilateral is $360°$. Thus, if $∠A = ∠C$ and…
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The Encyclopedia of Geometry (0214)
Problem The two diagonals of a rhombus are perpendicular to each other and bisect each vertex angle. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution $△ABD$ and $△CDB$ share $BD \ (=DB)$, and $$AB=CD \qquad and \qquad AD=CB,$$ $$∴ \quad…
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The Encyclopedia of Geometry (0213)
Problem The two diagonals of a rectangle are equal. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution For $△ABC$ and $△DCB$,$$AB=DC, \qquad BC=CB \qquad and \qquad ∠B=∠C \ (=∠R),$$$$∴ \quad △ABC≡△DCB,$$$$∴ \quad AC=DB.$$ $ $ $ $ $…
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The Encyclopedia of Geometry (0212)
Problem Within a parallelogram $ABCD$, draw $OA, \ OB, \ OC$, and $OD$ by connecting any point $O$ to each of the vertices, and let the midpoints of each be $E, \ F, \ G$, and $H$. Then, if the intersections of $DE$ and $CF$, $AF$ and $DG$, $AH$ and $BG$, and $BE$ and $CH$…
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The Encyclopedia of Geometry (0211)
Problem If equilateral triangles $ABD$ and $ACE$ are drawn outside a triangle $ABC$, and an equilateral triangle $BCF$ with the side $BC$ is drawn on the same side as a vertex $A$, then the quadrilateral $AEFD$ is a parallelogram. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$…
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The Encyclopedia of Geometry (0209)
Problem When you draw four squares with each side as one side on the outside of the parallelogram $ABCD$ and connect the centers of these squares to form a quadrilateral, it will be a square. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$…
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The Encyclopedia of Geometry (0210)
Problem If you draw the equilateral triangles $ABE$ and $CDF$ on the outside of a parallelogram $ABCD$ and the equilateral triangle $BCG$ on the same side as the parallelogram, then $$EG=AC \qquad and \qquad FG=DB.$$ $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$…
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The Encyclopedia of Geometry (0208)
Problem The sum of the lengths of the perpendicular lines drawn from the two vertices $A$ and $C$ of a parallelogram $ABCD$ to the line $XY$ outside the quadrilateral is equal to the sum of the lengths of the perpendicular lines drawn from the other two vertices $B$ and $D$ to $XY$. $$ $$ $$ $$ $\downarrow$…