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The Encyclopedia of Geometry (0184)
Problem When the bisectors of the four angles of a quadrilateral pass through the same point, the sum of the lengths of one pair of opposite sides of the quadrilateral is equal to the sum of the lengths of the other pair of opposite sides. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$…
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The Encyclopedia of Geometry (0183)
Problem If the longest side of a quadrilateral $ABCD$ is $AD$ and the shortest side is $BC$, then $∠BCD$ is greater than $∠BAD$ and $∠ABC$ is greater than $∠ADC$: $$∠BCD>∠BAD \qquad and \qquad ∠ABC>∠ADC.$$ $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$…
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The Encyclopedia of Geometry (0182)
Problem Let $ABCD$ be any quadrilateral, and let there be an interior point $O$ which is not the intersection of the diagonals; then the sum of $OA, \ OB, \ OC$ and $OD$ is greater than the sum of both diagonals $AC$ and $BD$: $$OA+OB+OC+OD＞AC+BD.$$ $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$…
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The Encyclopedia of Geometry (0181)
Problem In the following $(1)$ to $(4)$, for the condition on the left about a quadrilateral, if the condition on the right is a necessary and sufficient condition, answer $A$; if it is a necessary but not sufficient condition, answer $B$; if it is a sufficient but not necessary condition, answer $C$; if it is…
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The Encyclopedia of Geometry (0180)
Problem The perimeter of a quadrilateral is greater than the sum of its diagonals but less than twice the sum. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution For $△ABC$ and $△ACD$, $$AC<AB+BC,$$ $$AC<CD+DA,$$ $$BD<BC+CD,$$ $$BD<AB+DA,$$ $$∴ \ 2(AC+BD)<2(AB+BC+CD+DA),$$…
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The Encyclopedia of Geometry (0179)
Problem The sum of the interior angles of a quadrilateral is equal to four right angles. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution As shown in the diagram above, if you draw a diagonal $AC$ on a quadrilateral…
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The Encyclopedia of Geometry (0178)
Problem There is a circle inside a triangle. Prove that the perimeter of the triangle is greater than that of the circle. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution Create a triangle $A’B’C’$ that circumscribes the circle at…
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The Encyclopedia of Geometry (0177)
Problem Let $O$ be any point inside a triangle $ABC$, and the midpoints of $AO, \ BO$ and $CO$ be $L, \ M$ and $N$ respectively. Furthermore, if the midpoints of $BC, \ CA$ and $AB$ are $D, \ E$ and $F$ respectively, then the lines $DL, \ EM$ and $FN$ intersect at one point.…