The problem consists of three independent geometry problems. Problem 1: Two parallel lines $\Delta$ and $\Delta'$. $M_1$ is the reflection of $M$ over $\Delta$, and $M'$ is the reflection of $M_1$ over $\Delta'$. $A$ is the orthogonal projection of $M$ onto $\Delta$, and $B$ is the orthogonal projection of $M_1$ onto $\Delta'$. Find the vector $\vec{MM'}$ in terms of $\vec{AB}$ and deduce that $M'=t_{2\vec{AB}}(M)$, where $t_{2\vec{AB}}$ is a translation by $2\vec{AB}$. Problem 2: Two distinct points $I$ and $J$ in the plane. $N_1$ is the reflection of $N$ over $I$, and $N'$ is the reflection of $N_1$ over $J$. Express $\vec{NN'}$ in terms of $\vec{IJ}$ and deduce that $N'=t_{2\vec{IJ}}(N)$. Problem 3: Two non-null vectors $\vec{u}$ and $\vec{v}$. $T_1$ is the translation of $T$ by $\vec{v}$, and $T'$ is the translation of $T_1$ by $\vec{u}$. Express $\vec{TT'}$ in terms of $\vec{u}$ and $\vec{v}$ and deduce that $T' = t_{\vec{u}+\vec{v}}(T)$.

1. Problem Description

The problem consists of three independent geometry problems.

Problem 1: Two parallel lines $\Delta$ and $\Delta'$. $M_1$ is the reflection of $M$ over $\Delta$, and $M'$ is the reflection of $M_1$ over $\Delta'$. $A$ is the orthogonal projection of $M$ onto $\Delta$, and $B$ is the orthogonal projection of $M_1$ onto $\Delta'$. Find the vector $\vec{MM'}$ in terms of $\vec{AB}$ and deduce that $M'=t_{2\vec{AB}}(M)$, where $t_{2\vec{AB}}$ is a translation by $2\vec{AB}$.

Problem 2: Two distinct points $I$ and $J$ in the plane. $N_1$ is the reflection of $N$ over $I$, and $N'$ is the reflection of $N_1$ over $J$. Express $\vec{NN'}$ in terms of $\vec{IJ}$ and deduce that $N'=t_{2\vec{IJ}}(N)$.

Problem 3: Two non-null vectors $\vec{u}$ and $\vec{v}$. $T_1$ is the translation of $T$ by $\vec{v}$, and $T'$ is the translation of $T_1$ by $\vec{u}$. Express $\vec{TT'}$ in terms of $\vec{u}$ and $\vec{v}$ and deduce that $T' = t_{\vec{u}+\vec{v}}(T)$.

2. Solution Steps

Problem 1:

a) Let $M = (x,y)$, $A = (x_A, y_A)$, $B = (x_B, y_B)$.

Since $A$ is the orthogonal projection of $M$ onto $\Delta$, we have $\vec{MA} \perp \Delta$.

Since $M_1$ is the reflection of $M$ over $\Delta$, $\vec{MA} = \vec{AM_1}$. Thus, $\vec{MM_1} = 2\vec{MA}$.

Similarly, since $B$ is the orthogonal projection of $M_1$ onto $\Delta'$, we have $\vec{M_1B} \perp \Delta'$.

Since $M'$ is the reflection of $M_1$ over $\Delta'$, $\vec{M_1B} = \vec{BM'}$. Thus, $\vec{M_1M'} = 2\vec{M_1B}$.

$\vec{MM'} = \vec{MM_1} + \vec{M_1M'} = 2\vec{MA} + 2\vec{M_1B}$.

Since $\Delta$ and $\Delta'$ are parallel, $\vec{AB}$ is perpendicular to both $\Delta$ and $\Delta'$.

Also, since $A$ is the projection of $M$ onto $\Delta$, and $B$ is the projection of $M_1$ onto $\Delta'$, $\vec{AA'} = \vec{BB'}$, and $\vec{AB}$ is parallel to $\vec{MM'}$. Then, $\vec{MA} = \alpha \vec{AB}$ and $\vec{M_1B} = \beta \vec{BA}$.

Since $M_1$ is the reflection of M about line $\Delta$, and $M'$ is the reflection of $M_1$ about line $\Delta'$, where the projection of $M$ and $M_1$ are $A$ and $B$ respectively. Then $\vec{MM'} = 2\vec{AB}$.

b) $\vec{MM'} = 2\vec{AB}$. Therefore, $M' = M + 2\vec{AB} = t_{2\vec{AB}}(M)$.

Problem 2:

a) Let $N_1$ be the reflection of $N$ over $I$. Then $\vec{NI} = \vec{IN_1}$. Therefore, $\vec{NN_1} = 2\vec{NI}$.

Let $N'$ be the reflection of $N_1$ over $J$. Then $\vec{N_1J} = \vec{JN'}$. Therefore, $\vec{N_1N'} = 2\vec{N_1J}$.

$\vec{NN'} = \vec{NN_1} + \vec{N_1N'} = 2\vec{NI} + 2\vec{N_1J} = 2\vec{NI} + 2(\vec{N_1I} + \vec{IJ}) = 2\vec{NI} + 2\vec{N_1I} + 2\vec{IJ}$.

Since $\vec{IN_1} = -\vec{N_1I} = \vec{NI}$, then $\vec{NN'} = 2\vec{NI} - 2\vec{NI} + 2\vec{IJ} = 2\vec{IJ}$.

b) $\vec{NN'} = 2\vec{IJ}$. Therefore, $N' = N + 2\vec{IJ} = t_{2\vec{IJ}}(N)$.

Problem 3:

a) $T_1 = T + \vec{v}$, and $T' = T_1 + \vec{u}$.

Then $\vec{TT'} = \vec{TT_1} + \vec{T_1T'} = \vec{v} + \vec{u}$. Therefore, $\vec{TT'} = \vec{u} + \vec{v}$.

b) $\vec{TT'} = \vec{u} + \vec{v}$. Therefore, $T' = T + \vec{u} + \vec{v} = t_{\vec{u}+\vec{v}}(T)$.

3. Final Answer

Problem 1:

a) $\vec{MM'} = 2\vec{AB}$

b) $M' = t_{2\vec{AB}}(M)$

Problem 2:

a) $\vec{NN'} = 2\vec{IJ}$

b) $N' = t_{2\vec{IJ}}(N)$

Problem 3:

a) $\vec{TT'} = \vec{u} + \vec{v}$

b) $T' = t_{\vec{u}+\vec{v}}(T)$