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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.…
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The Encyclopedia of Geometry (0176)
Problem Let the midpoints of sides $BC, \ CA$ and $AB$ of a triangle $ABC$ be $D, \ E$ and $F$, respectively. Also, let $G$ and $H$ be the feet of perpendiculars drawn from $B$ and $C$ to any line passing through $A$, respectively, and $I$ be the intersection point of $EH$ and $FG$, or…
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The Encyclopedia of Geometry (0175)
Problem In a triangle $ABC$, suppose $AC>AB$. Let $D$ be a point on $CA$ such that $CD=AB$, $E$ be the midpoint of $AD$, $F$ be the midpoint of $BC$, and $G$ be the point where the extension of $FE$ intersects with the extension of $BA$, then $$AE=AG.$$ $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$…
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The Encyclopedia of Geometry (0174)
Problem There are two lines that intersect at a point $Q$. Now, on one of the lines, take three points $A, \ B$, and $C$ such that $QA=AB=BC$, and on the other line, take three points $L, \ M$, and $N$ such that $LQ=QM=MN$. Then, the three lines $AL, \ BN$, and $CM$ intersect at…
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The Encyclopedia of Geometry (0173)
Problem In a triangle $ABC$, let $AC>AB$. Let the perpendicular line from $B$ to $AC$ be $BH$. Let the perpendicular lines from a point $P$ on $BC$ to $AB$ and $AC$ be $PE$ and $PD$, respectively. Then $$PD+PE>BH.$$ $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$…
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The Encyclopedia of Geometry (0172)
Problem There are two half lines $OX$ and $OY$ starting at $O$. Let a point $P$ be within $∠XOY$ and the feet of perpendicular lines drawn from it to $OX$ and $OY$ be $Q$ and $S$, respectively. Then, if the difference between $PS$ and $PQ$ is a constant $m$, then the point $P$ is always…