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Problem 188

Given: Point \(P\) is in the interior of \(\triangle A B C\). Prove: $\mathrm{AP}+\mathrm{PB}+\mathrm{PC}>1 / 2(\mathrm{AB}+\mathrm{AC}+\mathrm{BC})$

Expert verified

We are given that point P is inside \(\triangle ABC\). To prove \(\mathrm{AP}+\mathrm{PB}+\mathrm{PC} > \frac{1}{2}(\mathrm{AB}+\mathrm{AC}+\mathrm{BC})\), apply the triangle inequality theorem to the smaller triangles \(\triangle APB\), \(\triangle APC\), and \(\triangle BPC\). Add the resulting inequalities, simplify, and divide by 2 to reach the desired result.

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Chapter 11

Point \(\mathrm{P}\) is in the exterior of \(\triangle \mathrm{ABC}\), in the opposite half plane of \(\mathrm{BC}\) from \(\mathrm{A}\), such that \(\mathrm{BP}=\mathrm{CP}\). $\mathrm{m} \angle \mathrm{ABC}>\mathrm{m} \angle \mathrm{ACB}$. Show \(\mathrm{m} \angle \mathrm{ABP}>\mathrm{m} \angle \mathrm{ACP}\).

Chapter 11

Given: \(\triangle \mathrm{ABC} ; \mathrm{D}\) is a point between \(\mathrm{A}\) and \(\mathrm{C} ; \mathrm{BD}>\mathrm{AB}\) Prove: \(\mathrm{BC}>\mathrm{AB}\)

Chapter 11

If the lengths of two sides of a triangle are 10 and 14, the length of the third side may be which of the following: (a) 2 (b) 4(c) 22 (d) 24 ?

Chapter 11

Given: \(\mathrm{AM}\) is the median of $\triangle \mathrm{ABC} ; \mathrm{m} \angle 2>\mathrm{m} \angle 1$. Prove: \(\mathrm{AC}>\mathrm{AB}\)

Chapter 11

Given: Quadrilateral \(\mathrm{ABCD}\) and straight rays \(\underline{\mathrm{ADF}}\) and \(\underline{\mathrm{ABE}}\). Prove: $\mathrm{m} \angle \mathrm{EBC}+\mathrm{m} \angle \mathrm{FDC}>1 / 2(\mathrm{~m} \angle \mathrm{A}+\mathrm{m} \angle \mathrm{C})$. (Hint: Draw \(\underline{\mathrm{AC}}\) )

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