Geometric Intuition for Caratheodory's Theorem (for Convex Sets) Consider the Wikipedia proof for Caratheodory's Theorem, the statement of which I have reproduced below.  In short, I am looking for some geometric intuition about the modified coefficients in the proof, something that I may have been able to "see" for myself if I were asked to prove the theorem without looking it up.

Theorem (Caratheodory).  Let $X \subset \mathbb{R}^d$.  Then each point of $\mathrm{conv}(X)$ can be written as a convex combination of at most $d+1$ points in $X$.

From the proof, each $y \in \mathrm{conv}(X)$ can be written as the following convex combination, where we assume $k \geq d+2$:
$$
y = \sum_{j=1}^k \lambda_j x_j
\text{ with } \sum_{j=1}^k \lambda_j = 1
\text{ and } \lambda_j > 0 \quad \forall\, j=1,\dots,k
$$
The resulting $k \geq d+2$ points $x_j \in \mathbb{R}^d$ are affinely dependent, so
$$
\sum_{j=1}^k \mu_j x_j = 0 \text{ with } \sum_{j=1}^k \mu_j = 0
$$
The remainder of the proof uses some funky manipulations of the coefficients for $y$ to show that one of the points in the convex combination for $y$ is really unnecessary.  The new coefficients are:
$$
y = \sum_{j=1}^k \left(\lambda_j - \frac{\lambda_i}{\mu_i} \mu_j \right) x_j
$$
where $i = \arg\min_{j \;:\; \mu_j > 0} \frac{\lambda_j}{\mu_k}$.  The $i$th coefficient turns out to be zero, completing the proof.  I understand why this choice of coefficients is desirable, but I do not understand why it's the "right" or "obvious" choice.  My own drawings do not make the situation any clearer to me.
What do the new coefficients mean geometrically, and in particular, how can I interpret the ratio $\lambda_i/\mu_i$ geometrically?  What does the $\max$ correspond to? 

You basically add
$$y = \sum_{j = 1}^k \lambda_j \, x_j$$
and
$$0 = \sum_{j = 1}^k \alpha \, \mu_j \, x_j, $$
for some $\alpha \in \mathbb{R}$. This yields
$$y = \sum_{j = 1}^k \underbrace{(\lambda_j + \alpha \, \mu_j)}_{=:\Lambda_j} \, x_j. $$
This directly yields
$$\sum_{j=1}^k \Lambda_j = 1.$$
However, you additionally need
$$\Lambda_j \ge 0 \;\forall j \qquad\text{and}\qquad \Lambda_i = 0 \text{ for some } i,$$
such that you obtain a convex combination, in which one coefficient is zero.
Now, try to figure out how to choose $\alpha$ and $i$.
