document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1166052>Question 1166052</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Let ABCD be a parallelogram and P a point in its interior. Define points<BR>\n" );
document.write( "Q, R and S such that A is the midpoint of PQ, B is the midpoint of QR<BR>\n" );
document.write( "and C is the midpoint of RS. Prove that D is the midpoint of SP. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #852869 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(2141)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=852869\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> This is a great problem that can be solved elegantly using **vector algebra**.\r<BR>\n" );
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document.write( "## 📐 Proof using Vector Algebra\r<BR>\n" );
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document.write( "Let the origin be $O$. We will represent the position of each point using its position vector (e.g., the position vector of point $A$ is $\vec{a}$).\r<BR>\n" );
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document.write( "### 1. Parallelogram Property\r<BR>\n" );
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document.write( "Since $ABCD$ is a parallelogram, the sum of vectors from opposite vertices is equal.<BR>\n" );
document.write( "$$\vec{d} - \vec{a} = \vec{c} - \vec{b}$$<BR>\n" );
document.write( "Rearranging this gives the fundamental vector property of a parallelogram:<BR>\n" );
document.write( "$$\vec{a} + \vec{c} = \vec{b} + \vec{d} \quad \text{(Equation 1)}$$\r<BR>\n" );
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document.write( "---\r<BR>\n" );
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document.write( "### 2. Midpoint Conditions\r<BR>\n" );
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document.write( "The problem defines points $Q$, $R$, and $S$ based on midpoint conditions:\r<BR>\n" );
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document.write( "* **A is the midpoint of PQ:**<BR>\n" );
document.write( "    $$\vec{a} = \frac{\vec{p} + \vec{q}}{2} \implies \vec{q} = 2\vec{a} - \vec{p} \quad \text{(Equation 2)}$$<BR>\n" );
document.write( "* **B is the midpoint of QR:**<BR>\n" );
document.write( "    $$\vec{b} = \frac{\vec{q} + \vec{r}}{2} \implies \vec{r} = 2\vec{b} - \vec{q} \quad \text{(Equation 3)}$$<BR>\n" );
document.write( "* **C is the midpoint of RS:**<BR>\n" );
document.write( "    $$\vec{c} = \frac{\vec{r} + \vec{s}}{2} \implies \vec{s} = 2\vec{c} - \vec{r} \quad \text{(Equation 4)}$$\r<BR>\n" );
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document.write( "---\r<BR>\n" );
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document.write( "### 3. Substitution and Simplification\r<BR>\n" );
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document.write( "We will substitute Equation 2 into Equation 3, and then the result into Equation 4 to express $\vec{s}$ in terms of $\vec{a}$, $\vec{b}$, $\vec{c}$, and $\vec{p}$.\r<BR>\n" );
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document.write( "**Substitute $\vec{q}$ (Eq. 2) into $\vec{r}$ (Eq. 3):**<BR>\n" );
document.write( "$$\vec{r} = 2\vec{b} - (2\vec{a} - \vec{p})$$<BR>\n" );
document.write( "$$\vec{r} = 2\vec{b} - 2\vec{a} + \vec{p} \quad \text{(Equation 5)}$$\r<BR>\n" );
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document.write( "**Substitute $\vec{r}$ (Eq. 5) into $\vec{s}$ (Eq. 4):**<BR>\n" );
document.write( "$$\vec{s} = 2\vec{c} - (2\vec{b} - 2\vec{a} + \vec{p})$$<BR>\n" );
document.write( "$$\vec{s} = 2\vec{c} - 2\vec{b} + 2\vec{a} - \vec{p}$$<BR>\n" );
document.write( "Group the vectors using the parallelogram property (Equation 1).<BR>\n" );
document.write( "$$\vec{s} = 2(\vec{a} + \vec{c}) - 2\vec{b} - \vec{p}$$\r<BR>\n" );
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document.write( "**Now, substitute $\vec{a} + \vec{c} = \vec{b} + \vec{d}$ (from Eq. 1):**<BR>\n" );
document.write( "$$\vec{s} = 2(\vec{b} + \vec{d}) - 2\vec{b} - \vec{p}$$<BR>\n" );
document.write( "$$\vec{s} = 2\vec{b} + 2\vec{d} - 2\vec{b} - \vec{p}$$\r<BR>\n" );
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document.write( "The $2\vec{b}$ terms cancel out:<BR>\n" );
document.write( "$$\vec{s} = 2\vec{d} - \vec{p}$$\r<BR>\n" );
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document.write( "---\r<BR>\n" );
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document.write( "### 4. Conclusion\r<BR>\n" );
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document.write( "Rearrange the final expression to check the required midpoint condition:<BR>\n" );
document.write( "$$\vec{s} = 2\vec{d} - \vec{p}$$<BR>\n" );
document.write( "$$\vec{s} + \vec{p} = 2\vec{d}$$<BR>\n" );
document.write( "$$\vec{d} = \frac{\vec{p} + \vec{s}}{2}$$\r<BR>\n" );
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document.write( "This is the definition of the midpoint of $SP$. Therefore, **$D$ is the midpoint of $SP$**. </SPAN>\n" );
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