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The Encyclopedia of Geometry (0043)
Problem In $△ABC$, when $AC$ is larger than $AB$, $AD$ connecting any $D$ on $BC$ and $A$ is smaller than $AC$. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution If $AC>AB$, then from…
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The Encyclopedia of Geometry (0042)
Problem The sum of two sides of a triangle is greater than the third side, and the difference between the two sides is less than the third side. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$…
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The Encyclopedia of Geometry (0041)
Problem If two sides of one triangle are equal to two sides of another triangle, but the third sides are unequal, the angle opposite the larger side is greater than the angle opposite the smaller side. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$…
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The Encyclopedia of Geometry (0040)
Problem When two sides of one triangle are equal to two sides of another triangle, and the angles between the two sides are unequal, the side facing the larger angle is greater than the side facing the smaller angle. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$…
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The Encyclopedia of Geometry (0039)
Problem When the two angles of a triangle are unequal, the side opposite the larger angle is longer than the side opposite the smaller angle. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ SolutionIn…
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The Encyclopedia of Geometry (0038)
Problem When two sides of a triangle are unequal, the angle opposite the longer side is greater than the angle opposite the shorter side. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution Suppose…
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The Encyclopedia of Geometry (0037)
Problem Regarding two congruent triangles $△ABC$ and $△A’B’C’$, when the two pairs of sides $AB$ and $A’B’$ and $AC$ and $A’C’$ are perpendicular to each other, the remaining pair $BC$ and $B’C’$ are also perpendicular to each other. However, assume that $∠A$ and $∠A’$ are not right angles. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$…
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The Encyclopedia of Geometry (0036)
Problem There are $△ABC$ and $△A’B’C’$. $$AB=A’B’, \ BC=B’C’, \ and \ ∠A=∠A’.$$ Then, under which of the following conditions can we always have $△ABC≡△A’B’C’$? (1) Either $△ABC$ or $△A’B’C’$ is an obtuse triangle. (2) $∠C$ is a right angle. (3) $BC>AB$. (4) $AC$ and $A’C’$ are the maximum sides of $△ABC$ and $△A’B’C’$, respectively. (5) $AB>AC$…
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The Encyclopedia of Geometry (0035)
Problem When the two sides of one of two triangles are equal to the two sides of the other, and the angles to one set of equal sides are also equal, the angles to the other set of equal sides are equal or supplementary angles. In the former case, the two triangles are congruent. $$…
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The Encyclopedia of Geometry (0034)
Problem Two triangles which have two equal sides and equal median lines are congruent. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution Let $AB=DE$, $AC=DF$ and $AP=DQ$. If we take points $A’$ and…