Based on the example provided, your objective, and your preference to avoid using flexbox for compatibility with IE10, I suggest utilizing display: table for the container in question.
By employing this approach, you can utilize the vertical-align properties within the "table-cell".
Feel free to explore the demonstration below. I have included a toggle button for showcasing the visibility of your captcha. For the best experience, you may want to view it in full screen.
$("#toggle").on('click', function() {
$(".captcha").toggle();
});
html,body {
height: 100%;
}
.outer-wrapper {
height: 100%;
width: 100%;
background: #eeeeee;
display: table;
}
.inner-wrapper {
height: 100%;
width: 100%;
display: table-cell;
vertical-align: middle;
height: inherit;
}
.login-wrapper {
background-color: #172238;
color: white;
width: 200px;
text-align: center;
height: auto;
display: block;
margin: 0 auto;
}
.captcha{
margin-top: 250px;
display: content; // This one changes
}
.homologation-line {
min-width: 200px;
background-color: #ffd800;
color: black;
text-align: center;
height: 30px;
}
#toggle {
position: absolute;
}
<script src="https://ajax.googleapis.com/ajax/libs/jquery/2.1.1/jquery.min.js"></script>
<button id="toggle">toggle captcha</button>
<div class="outer-wrapper">
<div class="inner-wrapper">
<div class="login-wrapper">
<p>Login</p>
<div class="captcha">
ENTER THE DIGITS:
</div>
</div>
<div class="homologation-line">
HOMOLOGATION
</div>
</div>
</div>